Hi, I'm an LA native... born and raised...
Question - Are you bringing a car?
If no, then give Palms/ Venice a shot. Be careful, since some parts can be kind of seedy and have their share of gang activity, but on the whole you should be okay. Rent will still be pricey...
There are lots of Westsider snobs (hehe) on this site that will tell you to go live in Santa Monica or "west of the 405" because as far as their concerned, that's where life begins and ends. Just be forwarned that rent out there is pricey, and traffic SUCKS. I was on the 405 this past Saturday, and it took me 45 mintues to drive from the 101 freeway to the Santa Monica Blvd. exit (it's only a couple of miles over the hill). Totally ridiculous.... I guess the weather was too nice and everyone was out and about. I will not miss the insane LA traffic when I go to school out of state...
If you want to save some money and you will have a car, try heading east a little bit. I'm an advocate of eastside living, but since you're going to UCLA, I'd say you should not go east of Korea Town. And for the record, DO NOT live in Korea Town.. it's just as bad as Watts, but they haven't had a riot yet to put them on the map. Give the "Mid-City" area a shot, as well as West Hollywood. There's a lot of great apartments near Melrose Ave. area (Melrose and Fairfax) as well as around 3rd street (3rd and Fairfax, to La Cienega and 3rd). You might even find some good deals in Beverly Hills, believe it or not. And if you check out the homes in the neighborhoods off Beverly Blvd (Bet. Beverly/ La Brea to Beverly/ Wilton) you can find a lot of great deals on guesthomes/ back houses.
If you really want to save dinero, you can commute from the "Valley" (North Hollywood, Studio City, etc.) but I leave that up to you. Another option would be to live in Downtown LA... that would be going just east of the 110 freeway, and while this suggestion contradicts my earlier point, I'm only suggesting it in the event that you get an internship or other opportunity with a downtown law firm or courthouse... it would be great to be close by!
Also, give http://aptmag.com
a shot to get an idea of apts. in the area.
Good luck, and have fun at UCLA!