The answer is: Why would they? The November contest will only give her the seat for the next two years-she's finishing out Hillary Clinton's term. In 2012 she'll have to run for a full term and I suspect strongly that a prominent New Yorker (George Pataki, Rudy Guiliani or Peter King) will jump in. Pataki and Guiliani have 100% name ID and could raise large amounts of money. Certainly Gillibrand will have an advantage in 2012 because New York is such a blue state and key Democratic groups will come out to vote for their candidate for President (which, by the way may not be Barack Obama but Hillary Clinton-more on that later) but voters in that state will split their tickets. They voted for Bill Clinton in 1992 and, in the same election, reelected Republican Senator Al D'Amato. As the state turned deep blue in the late 1990's they elected GOP governor George Pakaki. I suspect strongly that the Republicans will look closely at this seat, especially if they see that this could be the crucial 60th GOP vote in the Senate in January, 2013. In that case they will beg Pataki or Guiliani to run. In the end, Guiliani is probably the better bet because it looks likely that Pataki is going to take a shot at the Presidency in 2012.
By the way, who is running for the GOP in 2012? Pataki looks to be in. Tim Pawlenty is probably going to jump in. Newt Gingrich is almost a lock to run. If his wife gives him the go ahead Mitch Daniels will run as well. Will Sarah Palin run? No-but she'll be a kingmaker during the primaries and at the convention.