I'd suggest something north of Downtown, close to Griffith Park and/or the LA River trail. It would be close enough to USC for a reasonable commute. Neighborhoods would be Echo Park, Atwater, Frog town, Silver Lake, Los Feliz. Another option might be South Pasadena/Pasadena - lots of good rides around there.

Avoid Glendale, though. For some reason, it attracts all the crazy drivers.