I thought I would complement this answer that is already quite good! The Deira Spice Souk is obviously an inescapable area in Dubai. You'll find there all you need to cook almost anything.
Then, as you said in your original message, dates are really good there.
Another thing that is typical of the area is the honey and especially the Yemeni honey. You can find some in Dubai Mall and I am sure you'll be delighted with it. It is among the best (and also most expensive) around the world. The taste is pretty strong and it is well known to have a lot of benefits for the healthcare.
Beside honey, spice and dates, you might want to bring back some local arab cookies. Those are delicious, made with honey, almonds, etc... All you need to have a good time :)
You can find more details about Dubai souvenirs in this post.