First names database design
Adding a surrogate id will not help in this case.  You already have a good PRIMARY KEY(name), correct?

The 4 columns look good for the purposes mentioned.  You probably need a composite INDEX(group_id, name) to facilitate finding all the related queries.

What do you do about nicknames that apply to two standard names?

What do you do about someone with a non-"standard" name as their 'official' name?