I play Throne Room on Band of Misfits and choose Feast. Since it left play, the next time I play Band of Misfits I can choose a new card, as per this ruling (from the Complete and All-Encompassing FAQ):
If you use Throne Room to play Band of Misfits twice, it will be the chosen card the second time you play it, so you will play it as the same card twice. However, if the card is removed from play (like a Feast), it will be just a Band of Misfits again the second time. You can then choose another Action card from Supply to play it as. The card will stay where it is, and the chosen card's abilities will trigger even as the Band of Misfits will immidiately revert back to being just Band of Misfits (since it's not in play), but you will resolve the triggered abilites of the card.
What if I choose Island for the second use? I don't think Band of Misfits has lost track of itself, so it seems like it should be able to move itself from the trash to your Island mat. Is that correct?