The wikia for the issue names the character as "The Mystery Anthropoid" in the characters section and has the following to say about who he could be.
The identity of the Mystery Anthropoid has been much discussed. Several possible true identites have been suggested, from Sam Simeon, to Big Bear of the Forever People, to Highfather himself.
The following CBR articles also suggest some possibilities.
The person with the ape hands in this issue is never explicitly named, although the best theory I’ve heard is the typewriting monkey from Grant Morrison’s Animal Man run. Note that the monkey is even referred to in Superman Beyond. And like the typewriting monkey, the character in Final Crisis #5 knows a whole lot about the power of stories. Heck, this very well could be the origin of the monkey (a being experimented on in Command-D)! But really, that’s just a theory. It is mostly a mystery who the ape hands man is.
CBR, Final Crisis “Cliffs Notes”
Who was that ape man that was imprisoned with Nix Uotan and the rubicks cube solver (Metron?) in Darkseid’s lair?
One interesting answer I’ve heard was an experimented on Himon, as Himon was the creator of the motherbox, so it would follow that he would create the motherboxxx that was contained in the Rubik’s Cube, no?
However, Morrison I believe has stated that no New Gods survived (until the Fifth World, of course), so that likely rules Himon out.
That leaves the type-writing monkey from Animal Man as the leader, which is possible enough, as the monkey sure would know about the power of stories, wouldn’t he?
Wouldn’t it be a kick if this was the “origin” of the typewriting monkey from Animal Man? That he was created in Command-D!
In any event, basically it is unknown for sure.
CBR, Final Crisis FAQ
However, these are all just theories and it appears the character has yet to be named.