do sth to one's hearts content -> doesn't mean anything as ungrammatical. do sth to one's heart's content -> do something to the content of the heart of one

"hearts" has no apostrophe, so it can only mean "more than one heart." An apostrophe indicates either possession or contraction. So "heart's" can mean either "belonging to a/the heart" or "heart is" or "heart was", etc. However, these days things like "do something to one's hearts content" is becoming increasingly prevalent. The apostrophe is being threatened with extinction due to the destruction of its natural habitat. :)

The second one!