Choose the responsibility as System Administrator,
1. Navigate to Security -> User -> Define.
2. Search for the user for which you want to add new responsibility.
3. Bottom page shows the current responsibilities attached to current user. There you create a new record with the desired responsibility and then save.
4. Logout of the Apps. And login after sometime as most cases, it take some time to update its cache.
5. Next time, when you login with that user, you can see that new responsibility in your welcome page.