“Lite guv” (or “lite gov”) is an abbreviation for “lieutenant governor.” Texas political pundit Molly Ivins (1944-2007) used “lite guv” in many of her writings, applying the term since at least 1987.
 
The term “lite guv” is used in other states, but not as frequently as in Texas.
 
 
Wikipedia: Lieutenant Governor of Texas
The Lieutenant Governor of Texas is the third-highest executive office in the government of Texas, a state in the United States of America. It is the most powerful post in Texas government because its occupant controls the work of the Texas Senate and controls the budgeting process as a leader of the Legislative Budget Board. The Lieutenant Governor assumes the powers of Governor when the Governor is out of the state or otherwise unable to discharge the office. The Lieutenant Governor is elected separately from the Governor, rather than on the same ticket; it is therefore possible for the Governor and Lieutenant Governor to be from different political parties (which was the case during Governor George W. Bush’s first term). The Lieutenant Governor becomes Governor if the elected Governor resigns or dies while in office (current Governor Rick Perry took office upon George W. Bush’s ascendance to the office of President of the United States) in 2001.
 
Under the provisions of the Texas constitution, the Lieutenant Governor is President of the Texas Senate. By the rules of the Senate, the Lieutenant Governor establishes all special and standing committees, appoints all chairpersons and members, and assigns all Senate legislation to the committee of his choice.
 
The Lieutenant Governor decides all questions of parliamentary procedure in the Senate. He or she also has broad discretion in following Senate procedural rules.
 
The Lieutenant Governor is an ex officio member of several statutory bodies. These include the Legislative Budget Board, the Legislative Council, the Legislative Audit Committee, the Legislative Board and Legislative Council, which have considerable sway over state programs, the budget and policy. The Lieutenant Governor is also a member of the Legislative Redistricting Board (together with the Speaker of the House, Attorney General, Comptroller, and Land Commissioner), which is charged with adopting a redistricting plan for the Texas House of Representatives, Texas Senate, or U.S. House of Representatives after the decennial census if the Legislature fails to do so.
 
In the case of a vacancy in the Lieutenant Governor’s office, the Senate elects one of its members to act as President of the Senate until the next statewide office election, in effect becoming the Lieutenant Governor. A Senator elected as presiding officer in this way retains his district seat and the voting privileges entailed with his Senate election.
 
15 August 1987, The Nation, “The Lege Has a Taxing Session” by Molly Ivins:
Lite Guv. Do they have this much fun in Nebraska? In late April, the Governor went on a seventeen-city tour of Texas to convince the citizenry we didn’t…
       
23 December 1990, Kerrville (TX) Times, “Capital Dispatches” by Edd O’Donnell, pg. 4A, col. 4:
On the Senate side, Lite Guv Bob Bullock will be trying to pull his people together, clean the Republicans out of chairmanships an figure out how he can get anything passed with the House in an uproar.
 
Google Books
Molly Ivins can’t say that, can she?
By Molly Ivins
New York, NY: Vintage Books
1992
Pg. 276:
“Lite Guv,” all her friends said. “Go for Lite Guv.” Trouble with Lite Guv is, it pays squat: $7200 a year, just like the members of the Legislature.
   
The Texas Tribune
2010: Ronnie Earle Files for Lite Guv
by Reeve Hamilton
December 18, 2009
Former Travis County District Attorney Ronnie Earle was expected to file for some statewide race, though precisely which one has been something of a mystery.  Governor and attorney general have both been tossed around as possibilities.  Now, we know — he wants to be the next lieutenant governor.
 
Austin (TX) American-Statesman
At mayhem central, we relish team of Hutchison and Earle
Ken Herman, AUSTIN AMERICAN-STATESMAN
Published: 6:10 p.m. Monday, Dec. 21, 2009
(...)
In Texas, lieutenant governors (lite guvs to cool guys at the Capitol) are powerful due to the Senate president role that allows them to appoint committee chairs and hold great sway over the legislative flow.