I'm making my friend a tophat cake for his birthday. What should the base be?!

I am reading your question as: you are not asking how to make the brim of the hat but should you put the completed top hat on top of another cake or directly on the cake board. I think the answer to that would depend on how much cake you need. If the cake in the top hat will serve all of the guests then I would put it directly on the cake board. If you are going to need more cake than what the hat is...then I would make a square cake to put the hat on.
Either way is going to look great.

I'd get a dark chocolate pastry crust- lke a dark chcolate graham cracker crust.
OR make one using oreo cookies- it's good doing it this way, because they're dark ( almost top hat looking color)
make the base using ground oreos, some dark cocoa powder, flour, sugar etc.
Make an actual pastry base, bake it a little,

But if it's going to be the bottom of the cake it might not have to be super robust.

you could use a cardboard or you could make a cake cut the centre out and insert the top inside then frost it ,this way you would have 2 cakes but I would cut the base in half in order to make it rather low/flat after all the rim of the hat should be thin. I would use the cardboard and frost it instead.b

I would make it out of another sized cake. Just don't make it too thick or it won't look like the brim of a hat.