I think a straight forward puzzle game would be best for you two. Unfortunately I don't know of any multiplayer puzzle games. Perhaps a single player game that doesn't have a time limit, so you can stop and argue together over how to proceed:
"now what do we do here... oh I know we have to do this... no no that won't work, let's do this instead... but I don't want to do that... so do it your way and lose... I'm not going to lose... oh I lost... It's all your fault we lost you should have been more convincing..."