According to the Black Riders rules, Setup instructions on Player cards happen before you draw your first hand. The Setup instructions on the Ho! Tom Bombadil! boon instruct you to add the card to your hand. Does that mean that if I mulligan my first draw, I have to shuffle Ho! Tom Bombadil! into my decK?

I would just leave Ho! Tom Bombadil! in my hand even if I take a mulligan. I don't really have hard support for that. Here's what the core set manual says about your setup hand and mulligans:

"Each player draws 6 cards from the top of his player deck. If a player does not wish to keep his starting hand, he may take a single mulligan, by shuffling these 6 cards back into his deck and drawing 6 new cards."

It specifically says "6 cards." There are some cards that will override this, like the Gandalf's Delay burden card, but Ho! Tom Bombadil! doesn't seem to mess with that. So consider the "these 6 cards" in the core set manual text as referring to only the 6 cards that you drew for your setup hand, perhaps.

I've always held onto Ho! Tom Bombadil! regardless of mulligans, simply because it would seem stupid otherwise.

I have still ended up shuffling it away again on occasion by using Old Bogey-Stories.

I asked questions to Caleb about setting up cards in campaign mode, and this was one of them. Waiting for confirmation.