If one has the camera on their person, having it stolen shouldn't be too much of a worry. Best to use a wrist strap too.
As far as the $350 limit goes, that's going to hinder you a bit. Let's see what I can recommend as far as point-and-shoots go...
If you can push that budget a bit, go with a Canon S90
. It's a prime point-and-shoot, and to be honest I can't think of one better. If I had the money, I'd have one of these for times when I wouldn't be able to have my SLR with me.
If that doesn't work, there's also the Canon SX200 IS
. I've used this camera once before, at a KISS concert when security told me I couldn't use my SLR-- very different conditions to a convention, but I assure you it's damn sharp and a good size if one needs a good, compact camera. It's no S90, but it'll still do the job well.
Whatever one decides on, they should learn how to use the manual settings for shooting instead of relying on Auto. It'll result in much better photos if you have the patience for it, and it'll be good to know those things should one ever upgrade to a better camera like an SLR.