I must say, I really like this card. Enough that if I have a unique Hobbit and , it'll probably be an auto-include.
Basically, if you control a unique Hobbit, it serves as a mid-game mulligan, as the 1 extra card draw replaces the card itself. So with 3 of these in your deck, it essentially tightens it to a 47 card deck... assuming you're OK doing mid-game mulligans. The drawback is you may already have the cards you want, but simply can't pay for them yet, in which case, Drinking Song is taking up space until you can play it later. So where might it fit well? In decks with several duplicates that may clog your hand, and/or decks with various key cards you want to fish for, and you've no Daeron's Runes at the moment.
Without a unique Hobbit, it's harder to justify (but not impossible) as it no longer replaces itself. Still, in decks with large card draws, it'd be handy for those times when you're holding 10+ cards, but you still can't draw the 1 key card that's been eluding you, so it's time to "roll the dice" and see if it pops up.
I love the idea and artwork, which hints at Hobbits being lax in their vigilance, with the result being you won't quite know what'll happen. Will your replacement hand end up worse? Or turn out better, such as when Pippin gazed into the Palantir?
One of the few cards in the game that I'd truly call a "fun" card, for its name, artwork, and an ability that provides a mid-game mulligan.