What neighborhood can you get the most for your money in Dubai?

Creekside or Tecom area.  The latter is close to Jumeirah and yet prices are lower for hotels.

The Creek area or Bur Dubai is an interesting place to stay and there are some great little hotels along the Creek such as Barjeel Heritage Guest House that you could check out. The area is more authentically Middle Eastern and you are still pretty close to all the action in Downtown Dubai.

Al Barsha and Tecom are two other good places to look for hotels with a slimmer budget and are also easy to move around from. Both areas have Dubai Metro stations in the vicinity and are relatively easy to reach Jumeirah or Dubai Marina and even Downtown Dubai.

If you are prepared to drive around a bit (which is not really avoidable in Dubai) these are great places to find reasonably-priced hotels.