Thanks for the input everyone, here's what I ultimately went with. From my house rules:
"Intelligence determines how many magic items your character may have attuned at any one time, as described below:
-- Characters with Intelligence 9 or less may attune 2 magic items (one fewer than described in the PHB).
-- Characters with Intelligence 10-15 may attune 3 magic items (as described in the PHB).
-- Characters with Intelligence 16+ may attune 4 magic items (one more than described in the PHB)."
I put a lot of thought into it.
I went with the +1/-1 model for magic item attunement because it worked in two directions: it gave a bonus for having a high Int as well as a penalty for having a low Int. I set the requirements high enough to avoid cherry-picking, but low enough to make them attainable-yet-difficult: mid-level characters will have to struggle over whether or not to take feats, boost their primary stat, or boost their Int to gain one extra attunement. And lastly, I capped the bonus/penalty at 1 point to avoid The Wizard Problem, as well as magic item exploitation--intelligence boosting magic items aren't going to help you gain attunements.
Anyway. It's working like a charm so far, or at least as intended. We rolled up new characters over the weekend (using the Point Buy method) and only one character elected to start the game with minimum points in Intelligence. Instead of having a whole party of simpletons or a whole party of geniuses, I've got:
A human monk (Int 9)
A rock gnome rogue (Int 16)
A tiefling warlock (Int 11)
A human fighter (Int 14)
A wood elf cleric (Int 10)
Pretty sure that fighter is setting himself up for the Eldritch Knight path, and the rogue might be gunning for Arcane Trickster. Which is cool, because we've never had either in the party before. I'll keep you guys posted on whether or not the attunement tweaks cause problems down the road.