What access right is required to rename a computer

There are two things that are required - the domain account used must have the "Rename a Computer Account" right for the specific AD account and you have to have an account that is a member of the local administrators account on the computer being renamed.

By default Account Operators and Domain Admins have the Active Directory rights necessary to do this on computer objects but the right can be granted to any account if you wish to delegate management of specific computer objects to specific users or group(s).

When renaming computer accounts it's always a good idea to have a (tested and known to be good) local admin account just in case something goes wrong. It's important to realize that there are two different permissions in separate locations involved here - the Account Operators group will not automatically have local admin rights on the computer itself but Domain Admins generally will as they are (by default) added to the local administrators group on all computers in a domain.

The specific rights are detailed here and the default AD Groups and the permissions they are granted are detailed here.