Q: How to create a permission level that allows its members to manage group memberships? In a nutshell, I want to allow person X to add/remove people from any of the user groups we have created.
When adding a new permission level I can see a Site Permission described as "Manage Permissions  -  Create and change permission levels on the Web site and assign permissions to users and groups"
We don't want person X to create and change permission levels. Actually, we probably don't even want them to assign permissions to users or groups. We really only want them to add/remove users from existing groups.
How can I set that up?

A: You could do this by changing the group owner to person X. This is done by going to the group and going to Settings -> Edit group. In there set the owner to person X and he/she can edit the membership.

A: Although Eric's answer was close, we still didn't want to let "membership managers" have the ability to delete sharepoint groups.
As such, our solution was to push the functionality out to Active Directory. In this scenario we create a user group in Active Directory and place the AD user accounts (that should have permission) into that AD group. In Sharepoint we then add the AD group as the sole member of the SP group. 
This means staff with access to manage user accounts in AD can now add/remove members from the AD group and this will effect the permission changes in Sharepoint.
In one way this is still allowing greater permission than just managing group memberships, because in having access to AD, our staff has access to more than just managing group memberships, but the distinction is that granting a comparable level of admin rights within SP grants more than just user management ability - it would allow the admins to carry out functional changes to the site; in this case, the removal of user groups (at a minimum). 
As such, our final choice is actually a workaround to the original question.
