If you make, say, a Val or a Voiceless One who was both Psionic and Eldritch/Elder, could you apply the Arcane variation twice?

Sure. Psionics plays nice with other arcane talents.

From a spell perspective it makes sense, but could I give them Exceptional Specimen twice? While the spells work, it is more a "can you add the same variation twice" question.

What exactly do you mean by Exceptional Specimen? I don't see a spell or talent named that.

I'm pretty sure you can't add the same variation twice.

Exceptional Specimen is a Variation for threat statblocks. I figure you can't add it twice, but if they allow Arcane to be added twice for Psionic and Eldritch, by extension this should also be allowed twice. Hence my confusion.

Yeah, that's a more interesting question. I would say no, you can't add exceptional specimen twice (because its the same template). While arcane is also a template you can add to a creature (and technically you shouldn't be able to add it twice) my personal belief and preference is arcane (psionic) and arcane (eldritch/elder/primal...) should be two different things. I might eyeball the bonus to the arcanum skill if I ever did write a mod and "double dip" on the template. It's worth +1/2 a tier. So twice would be +1 to tier which should also mean +3 total to hit. Plus or minus a factor for any other weird/cool features (actually the arcanum template is pretty vanilla so that really doesn't apply).

I think anybody who spends time building monsters or characters knows that there is art as well as science to creation. For any given tier, a set of stat blocks could be easier (no ranged ability, no spell ability, no crazy talents) or harder (threat has spell casting, or useful bloodline talents, or Smite or Leadership or Murderous Precision or all the above). A mod writer has a duty to be "fair-ish" in designing threats. If the story needs a quirk or a twist though, its my opinion a mod writer is free to bend the rules somewhat (you know ... "fair-ish-ly")