False
Summary: She did sponsor a bill to restrict flag burning, though she technically wasn't the one who introduced it. But she also sponsored and/or introduced many other bills during her eight years in the Senate, possibly up to several hundred of them.
It is true that Clinton was a sponsor of the Flag Protection Act of 2005 (S. 1911, 109th Congress, which would have made it a crime to damage or destroy a flag "with the primary purpose and intent to incite or produce imminent violence or a breach of the peace, and under circumstances in which the person knows that it is reasonably likely to produce imminent violence or a breach of the peace", or to "intentionally threaten or intimidate any person or group of persons" by burning a US flag. She was not the one to actually introduce the bill; it was introduced by Utah Senator Robert Bennett, "for himself and Mrs. Clinton". The bill died in committee and never received a floor vote. (Thanks to Oddthinking for pointing out this bill.)
But it is certainly not true that this was the only bill that she introduced.
The independent non-partisan site govtrack.us has an option to search for bills for which she was a primary sponsor, and it lists 417 of them. If you click the text of each bill, there is a heading showing which Senator formally introduced it. The first several I checked were introduced either by her alone, or by her on behalf of herself and one or more other Senators. The list might also include some bills introduced by other Senators on her behalf, as was the case for the Flag Protection Act; I am not sure of the precise criteria used by govtrack.
That count does not include a further 1,950 bills of which she was a co-sponsor, which she would not have introduced herself.
If we narrow it down to bills introduced by her that were actually enacted into law, either by themselves, or as the corresponding bill from the House of Representatives, or as amendments to other bills, we have the following list:
S. 3625 (110th): A bill to designate the facility of the United States Postal Service located at 245 North Main Street in New City, New York, as the “Kenneth Peter Zebrowski Post Office Building”. Introduced by Mrs. Clinton (for herself and Mr. Schumer).
S. 3317 (110th): A bill to designate the facility of the United States Postal Service located at 101 West Main Street in Waterville, New York, as the “Corporal John P. Sigsbee Post Office”. Introduced by Mrs. Clinton.
S. 1148 (110th): Hudson-Fulton-Champlain Quadricentennial Commemoration Commission Act of 2007. Introduced by Mrs. Clinton (for herself, Mr. Schumer, Mr. Leahy, and Mr. Sanders).
S. 993 (110th): Pediatric Research Improvement Act. Introduced by Mrs. Clinton (for herself and Mr. Dodd).
S. 3847 (109th): A bill to designate the facility of the United States Postal Service located at 110 Cooper Street in Babylon, New York, as the “Jacob Samuel Fletcher Post Office Building”. Introduced by Mrs. Clinton.
S. 3716 (109th): A bill to designate the facility of the United States Postal Service located at 100 Pitcher Street in Utica, New York, as the “Captain George A. Wood Post Office Building”. Introduced by Mrs. Clinton.
S. 3613 (109th): A bill to designate the facility of the United States Postal Service located at 2951 New York Highway 43 in Averill Park, New York, as the “Major George Quamo Post Office Building”. Introduced by Mrs. Clinton.
S. 2722 (109th): A bill to designate the facility of the United States Postal Service located at 170 East Main Street in Patchogue, New York, as the “Lieutenant Michael P. Murphy Post Office Building”. Introduced by Mrs. Clinton (for herself and Mr. Schumer).
S. 2376 (109th): A bill to designate the facility of the United States Postal Service located at 80 Killian Road in Massapequa, New York, as the “Gerard A. Fiorenza Post Office Building”. Introduced by Mrs. Clinton (for herself and Mr. Schumer).
S. 272 (109th): Caribbean National Forest Act of 2005. Introduced by Mrs. Clinton (for herself and Mr. Schumer).
S. 1425 (108th): A bill to amend the Safe Drinking Water Act to reauthorize the New York City Watershed Protection Program. Introduced by Mrs. CLINTON.
S. 1241 (108th): Kate Mullany National Historic Site Act. Introduced by Mrs. CLINTON (for herself and Mr. SCHUMER).
S. 2918 (107th): A bill to designate the facility of the United States Postal Service located at 380 Main Street in Farmingdale, New York, as the “Peter J. Ganci, Jr. Post Office Building”. Introduced by Mrs. CLINTON (for herself and Mr. SCHUMER).
S. 1892 (107th): A bill to designate the facility of the United States Postal Service located at 375 Carlls Path in Deer Park, New York, as the “Raymond M. Downey Post Office Building”. Introduced by Mrs. CLINTON (for herself and Mr. SCHUMER) .
S. 584 (107th): A bill to designate the United States courthouse located at 40 Centre Street in New York, New York, as the “Thurgood Marshall United States Courthouse”. Introduced by Mrs. CLINTON (for herself, Mr. WELLSTONE, and Mr. DODD).
(It should be noted that the President throughout Clinton's service in the Senate was Republican George W. Bush, so that most major legislation enacted into law would have been introduced by Republicans. I suspect this accounts for the relatively trivial nature of the bills on this list.)