Not sure if this is a good bday cake but it's a fun cake like the dirt cake you mentioned.
Kitty Litter Cake....
Prepared a chocolate cake batter and poured half in one pan and half in another. Then I prepared a banana cake mix and poured it on top of the chocolate. Then I swirled them together. Baked them. While they were cooking, I took the sandwich cookies (like those hydrox cookies - I got the ones with a yellow top and a chocolate bottom - generic at Rainbow). I put the cookies in a food processor and made them into dust. Then I took some out and added green food coloring to it (to make it look darker like it had been peed on)
When the cakes were cool, remove them from the pan. I lined the pan with a plastic white small garbage bag. Then put the cake back in the pan. I frosted them (just used plain white frosting from a can). Then sprinkled them with the crushed cookies. Then sprinkled some green cookie crumbs on top of that.
Mircrowave some tootsie rolls and put them on top. Hang one over the edge to make it look like they missed.
Add a pooper scooper for the final effect.