You may not find this as a 100% satisfactory answer, but i'm going to give it a shot anyway. I have never attended UCLA, but will next year as a grad student and am currently in the process of house hunting. I did however, live in the Palms area a few years ago for about a year and a half when I worked in Century City, so I have a pretty good feel for the Westside neighborhoods. when I lived there I split a big 2br 2 ba with my best friend and it was about 1300. Prices have gone up aparently!
If you are looking for a good, chill area with coffee shops, decent walking traffic, and a good local life, i think you should look into Culver City. They have a nice little downtown and its an up and coming area. Santa Monica has some nice neighborhoody areas away from 3rd street, but you may find the area a little glitzy. Palms is okay, but a little boring. There isnt much there but a bunch of apartments, and a couple supermarkets. Also a bar with turtle races that i somehow over looked. And a great karaoke bar. Yeah, palms is alright. West LA isn't such a bad choice either.
You may want to avoid the east side of culver city,mar vista and venice, they are slightly dodgier then the other neighborhoods, but still not too bad. IT all depends on where you are coming from.