If a non-admin user wants to override Ansible variables, they only need to have the edit_ansible_variables permission.
However, if a non-admin user wants to update Ansible variables from the host details page, under the Ansible variables tab, they also need the edit_lookup_values permission.

Via API, CLI, ansible_variables edit page - only edit_ansible_variables is needed.
From the host details page, under the Ansible variables - also the edit_lookup_values is needed.