What is a "prize" for last place called?

I believe you are thinking of a consolation prize. It is a prize for consoling, or comforting, the loser by making him/her feel as if he/she still won something.

Merriam-Webster defines "booby prize" as "a prize that is given as a joke to the person who finishes last in a competition".

Consider "wooden spoon":

wooden spoon: a booby prize consisting of a spoon made of wood

(Princeton WordNet)

Wikipedia provides more information:

A wooden spoon is a spoon made from wood, usually given to an individual or team which has come last in a competition, but sometimes also to runners-up.

How about a "Red Lantern" award, for last place. In Alaska, it's a tradition in mushing, but the same award is given in other situations, such as the Tour de France, where it's the Lanterne Rouge. It comes from the red lantern on the caboose of a train.