Good question,
From what I know of the character of Elrond I do not think he would have. Firstly is he bound by the curse of Feanor (Honestly I cannot remember, I know that Galadriel was, but "Passed the Test" and so could go to Valinor)? Even if he is not then I guess that being of that kinship many would not have accepted him as king, even if he had single headedly taken the ring and threw it into mount doom after killing all nine nazgul and many other grand deeds that I cannot think of now (But this would make LotR a much less interesting story). I think the elves may have been a little sore still about Feanor and his whole affair.
I think that the elves may begrudge him the kingship, also I do not think Elrond would have wanted to BE king anyway. He'd probably find a house not unlike Rivendell and live there.
However that leads on to the question, IF he was offered the kingship and passed it by, would Elodan or Elhohir (Whichever was eldest) take the King ship? I think they would, Perhaps having seen Aragorn in his Kingly light may have made them fancy their hand at being a lord. Or perhaps I am going mad.
