Gold souk and shopping are excellent suggestions. Dubai is a pretty cool place, and even though it's been 5 or so years since I was there last I'm sure it's still great. The gold souk is in "downtown" Dubai near the creek. You could also get up to Sharjah to go to the rug souk. Lotsa fun.

There are numerous tour companies that I'm sure could arrange all sorts of desert tours, souk tours, Fujairah tours, oasis tours. Have fun! Maybe you can arrange a spa day the Burj.