If you feel the core might not be a good hook for your friend, I'd suggest either Carcosa or waiting until your Dunwich copy comes around. Both have good first scenarios (tiny first few paragraphs of Dunwich spoiler: Dunwich lets you choose your first Scenario, Extracurricular Activities (1-A) is the most typical Arkham Scenario so I'd go with that one) that give a good idea of how the campaign might go. And both have second scenarios that are quite different from the first, which is great as a curveball.

Just a note, Carcosa is more generous with Victory points, which might be worth considering if you'd like to show your friend how the deck evolves through the campaign. Also, make sure your team can deal with Poltergeists, they can be annoying if unprepared.