As with any city, the closer you get to downtown the more expensive things get. Not sure what your budget is, but you can look along I-10 just west of downtown in The Heights or Montrose area. The Montrose area is the gay/artistic/hippie part of town. Near the museums and such. Stay away from the east side of downtown unless you go way east, but that will defeat the purpose. There is always the Rice Village are near Rice University. Hella expensive, though. Minimum $1,800 a month for a one bedroom/studio set up.