What are some possible "functional" memory structures?
The topic is fairly broad. 

Functional hardware has been explored in e.g. Lisp machines.

More on the theoretical side, you may want to read about the various abstract machines which have been proposed for running functional languages.
Some of them are the SECD machine, the CAM (categorical), the Krivine machine, the STG machine.

Perhaps a more radical approach can be learnt by looking at the various graph-reduction machines. A pioneering work by Lamping describes how lambda can be run faster by rewriting graphs (!), instead of expressions.
More recently, linear logic simplified the several rewriting rules.

Some of these approaches can give you some hint at how a custom functional hardware should be made to speed up execution.

For conventional hardware, the STG machine seems one of the most promising approaches.