First, the easy part: the eidolon is simply "away" when the summoner is asleep. Nothing about the Leadership requires the leader to be available to his cohorts and followers 24/7.
While an eidolon technically can take the Leadership feat, his reduced hit dice (approximately 3/4 those of his summoner) and limited charisma (base 11, plus bonuses from evolutions, level increases, and magic items) would probably make the eidolon's cohort ineffective because of his low level relative to the summoner and thus to the PCs in the party.
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No, it's exactly as cheesy as giving a cohort leadership.
Here's the thing: Leadership requires that you have a Character Level of 7. And, well... Eidolons don't really get character levels.
Eidolons can't take Leadership because they never meet the pre-req of "Character level 7th". Unless I'm missing some text elsewhere, hit dice won't work for something so specific.