How much I would bet would depend almost entirely on the relative sizes of the players winnings up to that point. Given that only the winner gets to keep their winnings (the other two get nominal, fixed consolation prizes), the rational strategy is to do whatever gives you the best chance of coming out on top. There are a number of scenarios where the actual category might come into play, but for the most part it's the bankroll size that would determine my betting.
In an easy example, if you have more than twice as much as the next highest amount going into Final Jeopardy it would never make sense to bet any amount that would put you at risk if the player in second doubled up. Even if you were the world's leading expert in the category it would be irrational. So, if you have $20K and the next highest player has $7.5K, you should bet no more than $4,999. That would leave you with an unbeatable bankroll of $15,001 even if you got the answer wrong.