I was having this problem as well.
I had an access DB (MDB) that had link tables to a DB (source) that no longer existed. Also, the source db had an apostrophe in its name. Trying to delete the link gave the error specified in the question ('' is not a valid name...).
What I did:
Created a new MDB with the same name as the old source db (since the source no longer existed)
-Took the apostrophe out of the name
Added tables that reflected the name of the one linked in the troublesome database (the one that won't let you delete the links)
- Went to the troublesome database and right clicked a linked table and selected Linked Table Manager.
-Selected All and checked the box 'Always Prompt for New Location'
This asked me where it should look for the tables.
- Select the new db
- Make sure all tables are updated
- Go to troublesome database and right click and delete.
Some of these steps may be different from yours, depending on your exact situation.