For the Invite-a-Friend Program, does the person have to pay for TurboTax or can they file their taxes using the free option as well?

No, it does not apply to the free option.  For each qualified new-to-TurboTax Friend Referral (up to 50, max of $500), you get to choose a $10 Gift Card from over 30 companies and your Friend gets a special invitation for up to 20% off their federal return with TurboTax Online (those who select TurboTax Online Deluxe, Premier, and Self-Employed qualify for 20% off their federal return and those who select TurboTax Live Basic, Deluxe, Premier, and Self-Employed products qualify for 10% off their federal return. TurboTax CD/Download products do not qualify for this offer.)