To have “not one red cent” is to have absolutely no money. The U.S. penny (or “cent”) from 1793-1857 was made of reddish copper; the phrase “not worth a copper” was also popular in this period. “Red cent” has nothing to do with the Indian Head penny, made from 1859-1909.
 
The term “not a red cent” has been cited in print since at least 1837. The phrase is still sometimes used today, long after the “red cent” was manufactured.
 
     
Wikipedia: Penny (United States coin)
The United States one-cent coin, commonly known as a penny, is a unit of currency equaling one one-hundredth of a United States dollar. Its symbol is ¢. Its obverse has featured the profile of President Abraham Lincoln since 1909, the centennial of his birth. From 1959 (the sesquicentennial of Lincoln’s birth) to 2008, the reverse featured the Lincoln Memorial. 2009 will see several designs to honor Lincoln’s 200th birthday, while the coin will be re-designed in 2010. The coin is 0.75 inches (19.05 mm) in diameter and 0.061 inches (1.55 mm) in thickness.

The one-cent coin is often called a penny, but the U.S. Mint’s official name for this coin is “cent” and the U.S. Treasury’s official name is “one cent piece”.
 
History of composition
Years Material
1793–1857 copper
 
The Free Dictionary
red cent
n
(used with a negative) Informal chiefly US a cent considered as a trivial amount of money (esp in the phrases not have a red cent, not worth a red cent, etc.)
   
Merriam Webster’s Online Dictionary
Main Entry: red cent
Function: noun
Date: circa 1839
: penny —used for emphasis in negative constructions
   
(Oxford English Dictionary)
red cent n. orig. U.S. a one-cent coin (originally made of copper); (usually in extended use) a trivial amount of money; usu. in negative expressions.
1837 Spirit of Times 2 Dec. 333/3 Not a *red cent to my namenot even enough to buy a hoe-cake of a meal.
1889 Sir Charles Danvers xxix, I don’t care a red cent what you say.
1900 W. ARCHER Let. 1 Feb. in C. Archer William Archer (1931) xii. 263, I have never given a red cent for the ideas in plays.
1958 J. CAREW Black Midas ix. 193 He will pay you seven dollar..and not a red cent extra.
2003 K. KWEI-ARMAH Elmina’s Kitchen II. iii. 82 Tell them they can come burn down my place, before they get a red cent from me.
 
23 July 1839, Macon (GA) Weekly Telegraph, pg. 2:
Poor Mr. Clay, how we should pity him, with forty millions of money, and no friend base enough to touch “one red cent.”
   
16 October 1839, Milwaukee (WI) Daily Sentinel and Gazette, pg. 2, col. 2:
...neither did I ever receive from any person one red cent, for any political act with the rare exception that Charles James paid my expenses to the Rochester Convention,...
   
11 March 1841, Philadelphia (PA) Inquirer, pg. 2:
The sharper took him first to one place and then another, and finally into a coffee house, where he gave him a check on the Orleans Bank, which turned out not to be worth one red cent.
 
11 January 1844, Emanicpator (New York, NY), pg. 148:
...while some who signed the remonstrance ( and rich men too) gave not one red cent, after having promised to give several.