Most celebrities believe that there is no such thing as bad publicity. In other words, bad press is better than no press. And if you can get in the middle of a controversy, you are assured of attention. To a large extent, this need for publicity applies to politicians as well. Look at Katherine Harris. She was the Secretary of State for Florida during the 2000 presidential elections. After certifying that George W. Bush won the State of Florida by 537 votes, she was hated by Democrats (even receiving death threats) and loved by Republicans. She made the most of her publicity, writing a book Center of the Storm and running for Congress in 2002. She won.
Now that she has been in office only four months, she may be running for the Senate in the 2004 election (hat tip Taegan Goddard). Currently, she refuses to comment on it, but it makes sense - especially if Senator Graham does not run for re-election. Graham may decide to concentrate on the presidential election and he probably has the best chance of winning against Bush than any other Democrat in my opinion. The Democratic Party would not want Graham running for both seats (even if this is allowed under Florida law - I am not sure about this). If Graham won both positions, he would obviously be President and Florida Governor Jeb Bush would appoint a Republican to Graham's vacant Senate seat. So it is unlikely Graham will run for both positions, at least not if he can win the Democratic primary.
Let me explain a few things. 1) I am not predicting that Graham will win the Democratic primary. If he does, I think he has the best chance of beating Bush because he might carry Florida. However, there are others running that may be more appealing to Democrats 2) Graham could decide to run for both positions and see how the early primaries go. If he is doing well, he can drop out of the Senate race and let another Democrat run. If he is doing poorly, he could drop out of the Presidential race and stay in the Senate race. So I predict he will run for both offices.
However, it all depends upon Graham. Politics being politics, there are always options. If Graham would prefer a chance at the Presidency and would settle for being Vice-President (which is also a chance at the Presidency), then he will probably not run for the Senate. This is because of how important Florida is in the election. Even if Graham does not win the Democratic primary, he will probably be the winner’s first choice for vice-president because it might swing Florida.