I think cutting the grass, pulling some weeds and powerwashing that would make it look 10x better. Personally, i think before you go spend a bunch of money on a new deck/patio whatever, clean up what you got and see what you can do with it. You can see what could be a brick patio, but its covered in weeds. That and the other side is dirt... I worked in landscaping for a little over a year and you would be amazed how much better something can look just by pulling weeds, power washing the deck, cutting the grass then either laying down sod or making a flower bed or something.
EDIT: Personally btw, i like wood patios. I like being up off the ground even if it is a foot. Although my friend has a badass back yard, where he has a round brick patio thing, with a pond/fountain kind of incorperated in it. They had a square brick patio like yours, removed 1 side so it was like ) and put a little plastic pond thing there, then finishd around the pond with just a little brick, basically a trim. I could probably get him to send me a picture of it, he also has a fire pit thing on the brick, looks real good.