Yes! To do so, go to the Settings tab and tap your nickname in the "Family Members" list. You'll then have the option to change your nickname and your relationship.
Please note that changing your nickname or role in the album will not affect your permissions for the album. For example, if you were invited to the album as a grandparent or other relative, you'll still have the permissions associated with a grandparent or other relative even if you change your nickname and/or relationship to "Dad".