The city of Austin was named after Stephen F. Austin (1793-1836), known as the “Father of Texas.” A small village on the Colorado river was originally called Waterloo; it was selected as the capital city in 1839 and the name was changed to Austin.
 
The name of Austin for the capital city of the Republic of Texas was conceived before it was specifically applied to Waterloo. On January 14, 1839, a Texas commission was created to look for a proper site for the capital and “the name of said site shall be the city of Austin.” Two sites—one near the Brazos river and the other near the Colorado river—were under consideration.
 
The city of Austin might have been named by Anson Jones (1798-1858) of Brazoria, who would later become the fourth and last president of the Republic of Texas. The Telegraph and Texas Register (Houston, TX) of May 30, 1838 is the first known use of “city of Austin” in print:
 
“Mr. Jones of Brazoria introduced a bill to take the sense of the people on the location of the seat of Government at the city of Austin, which was ordered to be spread upon the journals.” 
 
 
Wikipedia: Stephen F. Austin
Stephen Fuller Austin (November 3, 1793 – December 27, 1836), known as the Father of Texas, led the second, but first legal and ultimately successful colonization of the region by bringing 300 families from the United States. The capital of Texas, Austin in Travis County, Austin County, Austin Bayou, Stephen F. Austin State University in Nacogdoches, Austin College in Sherman, as well as a number of K-12 schools are named in his honor.
 
Texas A&M University
Austin’s Colony
Austin’s Colony was the first legal settlement of North American families in Mexican-owned Texas. Led by the Empressario, Stephen F. Austin, an initial grant for three hundred families—the “Old 300”—in 1821 opened up Texas to a flood of American immigrants, as many as 30,000 by the time of the Texas Revolution in 1835. This colonial period that brought Anglo and African settlers from the United States into contact with the governmental and ranching traditions of Spain and Mexico helped set the course for much of Texas’ history in the 19th century—and was only overshadowed later by the discovery of oil in Texas in the 20th century.
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Austin next set out to find land for his colony. He decided on the rich river bottom between the Brazos and Colorado Rivers, south of the El Camino Real. The area of Austin’s Colony included land that is now contained in 19 Texas counties.
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Once in the colony, the settlers lined up at the land office in the colonial capital of San Felipe de Austin to receive their land. (...) San Felipe, capital of the colony, is located near Sealy along the Brazos River.
 
Wikipedia: Austin, Texas
Austin is the capital city of the U.S. state of Texas and the seat of Travis County. Located in Central Texas on the eastern edge of the American Southwest, it is the fourth-largest city in Texas and the 14th most populous city in the United States. It was the third-fastest-growing large city in the nation from 2000 to 2006. Austin has a population of 790,390 (2010 U.S. Census). The city is the cultural and economic center of the Austin–Round Rock–San Marcos metropolitan area, which has a population of over 1,716,291 (2010 U.S. Census), making it the 35th-largest metropolitan area in the United States.
 
The area was settled in the 1830s on the banks of the Colorado River by pioneers who named the village Waterloo. In 1839, Waterloo was chosen to become the capital of the newly independent Republic of Texas. The city was renamed after Stephen F. Austin, known as the father of Texas. The city grew throughout the 19th century and became a center for government and education with the construction of the Texas State Capitol and the University of Texas. After a lull in growth from the Great Depression, Austin resumed its development into a major city in the 1980s and emerged as a center for technology and business. Austin is home to many companies, high-tech and otherwise: Fortune 500 corporations Freescale Semiconductor, Forestar Group, and Whole Foods Market, are headquartered there; AMD, Apple, Broadcom, Google, IBM, Intel, Qualcomm, ShoreTel, Synopsys and Texas Instruments have prominent regional offices there. Also Dell’s Worldwide Headquarters is located in nearby Round Rock, a suburb of Austin.
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n 1835–1836, Texans fought for independence in the Texas Revolution and won. Texas thus became its own independent country with its own president, congress and monetary system. In 1839, the Texas Congress formed a commission to seek a site for a new capital to be named for Stephen F. Austin. Mirabeau B. Lamar, second president of the newly formed Republic of Texas, advised the commissioners to investigate the area named Waterloo, noting the area’s hills, waterways, and pleasant surroundings. Waterloo was selected and the name Austin was chosen as the town’s new name. The location was seen as a convenient crossroads for trade routes between Santa Fe and Galveston Bay, as well as routes between northern Mexico and the Red River. Austin is also the site where the southern leg of the Chisholm Trail leads to the Colorado River.
 
Handbook of Texas Online
WATERLOO, TEXAS (Travis County). Waterloo was on the north bank of the Colorado River approximately at the site of the Congress Avenue Bridge in Austin. Jacob Harrell a hunter who erected a tent on the river bank in 1835 and his family were early settlers in the area. He was visited in 1837 or 1838 by Mirabeau B. Lamar, who decided that the spot was an appropriate site for the capital city of the Republic of Texas. The town was surveyed by Edward Burleson in 1838. The five-man commission appointed in January 1839 to select the capital location was instructed by President Lamar to visit Harrell’s split-log stockade. The commission found four families living near the stockade and named the site Waterloo, the name under which it was incorporated when it was chosen to be the capital site. Congress designated the name of Austin for the new city, and the name of Waterloo was dropped. The land adjacent was relinquished by Logan Vandeveer, James Rogers, J. D. Hancock, J. W. Harrell, and Aaron B. Burleson. The most desirable spot in the 7,735-acre site was chosen for the capitol building. The new town was surveyed, and the first lots were sold in August 1839.
 
Handbook of Texas Online
AUSTIN, TEXAS (Travis County). Austin, the capital of Texas, county seat of Travis County, and home of the University of Texas at Austin, is located in central Travis County on the Colorado River and Interstate Highway 35. Situated at 30°16’ north latitude and 97°45’ west longitude, it is at the eastern edge of the Hill Country and the Edwards Plateau. The city was established by the three-year-old Republic of Texas in 1839 to serve as its permanent capital, and named in honor of the founder of Anglo-American Texas, Stephen F. Austin. A site-selection commission appointed by the Texas Congress in January 1839 chose a site on the western frontier, after viewing it at the instruction of President Mirabeau B. Lamar, a proponent of westward expansion who had visited the sparsely settled area in 1838. Impressed by its beauty, healthfulness, abundant natural resources, promise as an economic hub, and central location in Texas territory, the commission purchased 7,735 acres along the Colorado River comprising the hamlet of Waterloo and adjacent lands. Because the area’s remoteness from population centers and its vulnerability to attacks by Mexican troops and Indians displeased many Texans, Sam Houston among them, political opposition made Austin’s early years precarious ones.

Surveyors L. J. Pilie and Charles Schoolfield laid out the new town, working under the direction of Edwin Waller, who was appointed by Lamar to plan and construct Austin. Out of the 7,735 acres they chose a 640-acre site fronting on the Colorado River and nestled between Waller Creek on the east and Shoal Creek on the west. The plan was a grid, fourteen blocks square, bisected by Congress Avenue, and extending northward from the Colorado River to “Capitol Square.” Determined to have Austin ready by the time the Texas Congress convened in November 1839, Waller opted for temporary government buildings at temporary locations. The one-story frame capitol was set back from Congress Avenue on a hill at what is now the corner of Colorado and Eighth streets. The first auction of city lots took place on August 1. During October President Lamar arrived, government offices opened for business, Presbyterians organized the first church, and the Austin City Gazette, the city’s first newspaper, made its appearance. Congress convened in November, Austin was incorporated on December 27, and on January 13, 1840, Waller was elected the town’s first mayor. By 1840 Austin had 856 inhabitants, including 145 slaves as well as diplomatic representatives from France, England, and the United States.
   
Wikipedia: Anson Jones
Anson Jones (January 20, 1798 – January 9, 1858) was a doctor, businessman, congressman, the fourth and last President of the Republic of Texas, sometimes called the “Architect of Annexation.”
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Life in Texas
In 1833 Jones headed west to Texas, settling eventually in Brazoria.
 
The Portal to Texas History
30 May 1838, Telegraph and Texas Register (Houston, TX), pg. 1, col. 4:
ADJOURNED SESSION—2d CONGRESS.
REPORT OF THE PROCEEDINGS OF THE HOUSE OF REPRESENTATIVES.
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WEDNESDAY, May 28.
Mr. Jones of Brazoria introduced a bill to take the sense of the people on the location of the seat of Government at the city of Austin, which was ordered to be spread upon the journals.
 
The Committee on Enrolled Bills, reported the joint resolution for the benefit of Wm. Kuykendall.
 
The House adjourned till 3 o’clock, p.m.
 
The Portal to Texas History
9 January 1839, Telegraph and Texas Register (Houston, TX), pg. 2, col. 3:
Mr. Jenkins offered the following substitute, to wit:—“Two scites for the permanent location of the seat of government, one on or within ten miles of the Colorado river, and the other on or within ten miles of the Brazos, which scites, when so selected and reported to the President, shall on the——day of——be submitted for them to select between the two by general vote, and no vote given for any other place save the two put in nomination by said commissioners, shall be taken into consideration, or counted, ...”
 
[The Portal to Texas History
16 January 1839, Telegraph and Texas Register (Houston, TX), pg. 2, col. 5:]
[The Portal to Texas History
26 January 1839, Telegraph and Texas Register (Houston, TX), pg. 1, col. 1:]
The Portal to Texas History
Laws of the Republic of Texas,
Passed the First Session of Third Congress, 1839.
Pg. 161:
AN ACT
Entitled an act for the permanent location of the Seat of Government.
Sec. 1. Be it enacted by the Senate and House of Representatives of the Republic of Texas in Congress assembled, That there shall be and are hereby created five Commissioners, to be elected, two by the Senate and three by the House of Representatives, whose duty it shall be to select a site for the location of the Seat of Government, and that said site shall be selected at some point between the rivers Trinidad and Colorado, and above the old San Antonio Road.
Sec. 2. Be it further enacted, That the name of said site shall be the city of Austin.
Pg. 165:
JOHN M. HANSFORD,
Speaker of the House of Representatives.
DAVID G. BURNET,
President of the Senate.
Approved, January 14, 1839.
MIRABEAU B. LAMAR
   
The Portal to Texas History
1 May 1839, Telegraph and Texas Register, pg. 2, col. 4:
To the Editor of the Telegraph:
SIR:—I have seen an article in the “Morning Star,” which was transferred to your paper of the 17th inst., on the subject of the probable removal of the Seat of Government, and the new selection of the city of Austin.
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There are more people now living within twenty miles of the city of Austin, than there are within forty miles of the city of Houston.
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VERITAS.
 
The Portal to Texas History
29 May 1839, Telegraph and Texas Register (Houston, TX), pg. 3, col. 4:
NOTICE—The subscriber having been appointed the agent to lay off the new city of Austin and superintend the sales of lots, &c. &c., gives public notice that the commissioners appointed for that purpose, have selected the town of Waterloo on the Colorado river, about thirty-five miles above the town of Bastrop, to which point he will proceed forthwith to lay off a town, according to the provisions of an act entitled an act for the permanent location of the seat of government.
 
There will be a public sale of lots at said town on the first day of August next.
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EDWIN WALLER,
Agent.
City of Austin, April 22, 1839.
   
The Portal to Texas History
29 May 1839, Telegraph and Texas Register (Houston, TX), pg. 3, col. 5:
THE EPITOMIST—A weekly newspaper to be printed in the city of Austin, the new seat of government for the Republic of Texas, to be conducted by an association of literary gentlemen, and edited and published by James Burke.
 
The Portal to Texas History
4 November 1886, Austin (TX) Weekly Statesman, “Austin, the Capital of Texas,” pg. 4, col. 4:
The city was named after Stephen F. Austin, and its name was suggest by our present Gen. W. P. Hardeman’s father, Hon. T. J. Hardeman.