Maybe Henry got amnesia in the balloon crash, and didn't remember anything about himself. Not knowing your own name could be slightly maddening. In a desperate attempt to reclaim his identity, he took the name of the only person around. The dead Henry Gale.

Or maybe Henry is gay and was afraid the people on the Island were bigots, so he made up the whole wife thing, and took the name of his now dead boyfriend to honor him.

I'm going with the first theory though.