Found this item somewhere on the interwebs, I believe it was on a list of trinkets, oddities and magic items one might find in a hag’s lair.  I would like to incorporate it into our current campaign.

Lich’s Hand

A small coffer with a mummified hand of a lich with a sinew cord that allows it to be worn around the neck. A magic ring requiring attunement may be placed on the finger of the hand, and the wearer may utilize its abilities or benefits without counting against the number of attuned magic items allowed.

The item is cursed and the wearer will not willingly part with it. Additionally, the wearers alignment will slowly change to Evil if it is not already. The curse(and alignment change) can be broken/reversed with a Greater Restoration spell or a Wish.

Would you consider this a Rare or Very Rare item?

I don't see any benefit to the player, unless I'm missing something, so I don't think the rarity of this item really matters much. You probably want to give this out before Level 9, so a spell doesn't immediately remove it and they have to go on a quest to get the curse removed. You're probably more in the realm of Rare for items around that Tier (possibly even lower if you allowed the Level 3 Remove Curse to to remove the curse). In terms of the frequency of Liches in D&D lore, it seems like it would be Very Rare. Again, though - I don't think the rarity really matters for using it in your campaign.

I would recommend removing all the text about attunement from the item. Require attunement, but not taking up an attunement slot and providing no benefits upon attunement seems overly complicated when you could just be cursed upon getting the item and having nothing to do with attunement.

Read it again…The Lich’s Hand does NOT require attunement(although it is Cursed…and yes, there are official 5e cursed magic items that don’t require attunement such as the Loadstone).  It allows you to gain the benefit of a magic ring that requires attunement without using an attunement slot.  

Gotcha, I completely misunderstood that and thought you were stating that you can attune to the item you created (the Lich's Hand) without using an attunement slot. Hopefully I'm the only one silly enough to make that mistake as a hand is not a ring (switch the "maybe" to "may be" to make that a bit clearer). 

Now that I understand what the item actually does, I'd lean towards Very Rare. In reality, the item is so heavily dependent on the other items you give out as treasure that it's a bit murky, but gaining something akin to an extra attunement slot, even if it is only for rings, is such a powerful feature that I'd lean that way even if you don't end up giving out powerful rings. It lends more to the idea that Liches are rare and finding a magical, intact hand of theirs would be rare and powerful. 

One thing you might consider with this is what happens when the curse is broken. Normally, you'd still get the benefits, but given that the curse isn't mechanically disruptive (and the pace of the change seems to be up to the DM and player) and it makes sense thematically for the magic that provides the benefit to also be corrupting, I would suggest that if the curse is dispelled, the hand also loses its magic. This gives the item more use in higher tier play, so the cool flavor you have doesn't get washed away too quickly by the Greater Restoration spell.

For sure, the item is only of value to a character suffering from an attunement slot shortage who also possesses a magic ring requiring attunement.  In that case, it’s essentially an extra attunement slot, with, as you stated, a non-debilitating curse effect unless you are playing in a group of very Good aligned characters I suppose.