Aragorn did a few things in the books that can be attributed to his innate abilities as a Dunedain. For one thing he could heal not just regular ills but supernatural ills (healing hands of the king and all that, keeping in mind that the kings of Gondor were always Dunedain). For another he went toe to toe with Sauron through the palantir, and actually succeeded in the battle of wills that occured. There are references about how fearful Sauron would be if Aragorn used the One Ring, hinting that Aragorn has inner powers to be amplified much as Gandalf does. All of the Dunedain are supposed to be worth ten men in battle (or is it more?), which hints that as a group they are superior.
I think aasimar is the best idea, since in Middle Earth the high elves were basically all powerful celestial beings (Glorifindel glowing with light as he challenges the Dark Riders, for instance). If I were to put it in a game, I'd use a similar method to Mark's. The high born could become paladins, but this class would be restricted to them.