At what level have you seen parties typically begin to have to deal with "attunement overload" -- the situation where the party has so many magic items that require attunement that it cannot possibly have all of them attuned at the same time?
That in itself is not a bug, it's supposed to be a feature.
To be a problem, a player must have acquired more than three items of comparable strength and that it feels appropriate that all four (five etc) should be useable at the same time.
This simply does not happen very much if you run published modules - they have considerably less loot than either the random DMG tables yield or what prev-edition modules give out.
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In all the cases where the fourth item is clearly inferior to the other three it's an obvious solution to simply drop the weaker item. In all cases where the player agrees it might be a bit too much to be allowed all four powerful items at the same time, the rules work as intended.
I would say any houserule should aim for the remaining situations: such as when a character have brought along a family heirloom since perhaps level 1, and now at mid- to high level, that item can no longer compete for one of the three attunement slots. In this case the restriction is in the way of the story, and there is a real issue.
My suggestion is to allow a player to "split" one attunement slot into two low-power slot halves. That is, instead of attuning to one item of any power, you can use that slot to attune to two items of sufficiently lower power (as determined by player and GM).
Hint: Things that give universal bonuses (such as +1 to attacks, AC, saves etc) should never be considered "lesser", unless a very good reason (such as "using it is part of my identity" of family heirlooms above). The fact the bonus is "only" +1 instead of +2 or +3 is oterhwise not a good reason to allow it. Lesser items are items like Brooch of Shielding and Boots of Jumping that target low-level game effects (at higher levels Magic Missiles cease to be an issue, and people prefer flight/levitation/teleportation over mundane jumps and climbs). Another lesser item could be something that provides resistance to fire only, when other items start providing multiple benefits: resistance to A, B and C; or both resistance and +1 to AC for instance.
A player can then choose between attuning to 3 major items, or 6 minor items, or any combination thereof.