Other than judging the Merle Watson Banjo Contest every year, I'm kind of out of the banjo contest loop nowadays. There are major ones at Winfield, Kansas in mid-September and Topanga Canyon in southern California. Other than that, a lot of festivals do have contests, and will often list them in their ads. Most of those don't have big prizes such as famous brand banjos, but there may be a few hundred dollars in prize money. I don't know of any central source of information for banjo contests. One problem would be that they come and go, prizes change, etc.
As for how to increase your chances of winning, that may vary from place to place. In my judging I give a lot of credit to people with good command of Earl Scruggs' (or other masters') material, having obviously studied the originals. Then I listen for originality and interest value. I really like hearing good original tunes (and in fact, the contest rules for the Merlefest contests all give credit for that). Blistering speed is not important, but if the tune is well-executed and sounds good at high speed, that's impressive. Since you get to play more than one tune, having one that is a bit different, but sounds great, would have to be helpful.
When I had the opportunity to create the rules for a contest (Merle Watson Banjo Contest), I made sure to include an idea I'd never seen incorporated before: Having to play a *song* of someone else's choice and do all the breaks and backup, with almost no time to prepare. This tests a person's musicianship beyond being able to memorize and execute an exact pre-planned arrangement. Every year this contest pairs the finalists with a famous bluegrass singer/guitar player (like Del McCoury or Tim Stafford), and that is part of the judging, along with a tune of choice. It's the only contest I know of like that, but I hope it catches on.
Good luck with the contests, and be sure to pick it solid!