Thank you so much for all your suggestions and great feedback! I love the Dichondra idea though upon further investigation it seems that it is fairly difficult to start and maintain and it does need lots of water. Kind of the same thing I would be faced with going for lawn. But wow, it looks beautiful and lush, I love it!
I have been thinking about gravel and I do have some pathways in other areas that are treated this way but I can't bring myself to do it in such an expansive area. It would probably take a few truckloads.
I think I really want to go with some form of xeriscaping as long as it has a carpet feel and doesn't mind the kid roaming around in it on occasion. I have lush and mostly vertical landscaping around this area so I really would like to keep this clear and low and accessible.
Iceplant is a great idea. I am already using this inbetween some of my flowerbeds but till now didn't think of it as a lawn replacement. I haven't stepped on my flowerbeds much so I am not sure if it could take the occasional human traffic. Native plants sound great too, though it seems it's the native weeds that have been spreading fastest on that patch I am hoping to fix.
So looking at everything you guys suggested I am thinking all types of different creeping thyme might do the trick, you know: red, white, green, purple - the more colors the better. If it turns out thick enough it could look great between the palm trees and create a bit of a "What dreams may come" feel. Do you guys think this is doable? Will it spread across such a large area? Maybe if I do well-spaced patches of different thyme they will grow and connect and eventually form that carpet I am after. If there remain holes I could just try and fill them up with something else of a similar nature. I do believe there's a blue variety of Trailing Rosemary. Maybe even try creeping Phlox. Do you think this could work?
Perhaps three tenniscourts worth of this would become a bit too psychedelic?
Hmmm... I guess it's all about the size of the patches.
I do love the look of this Phlox:
I should follow your book suggestions before I lock in on this...