I don't know, then. I know people have had issues when trying to add a non-gmail account, but I've never tried adding a household member by selecting a contact. I suppose you could try typing in their email address.
It could be something wrong on Google Nest's servers, or a glitch in the Google Home app. If it were happening to me, I'd be tempted to just try deleting and reinstalling the Google Home app. I've done that on my iPhone to resolve a completely different issue.
Google Nest Support: You really should document that, in order to be invited to be a member of a Google Nest household, the prospective member has to have a Google (gmail) account and have the Google Home app installed. And appropriate error messages should be displayed in the app. And Support should tell customers this when they post about the problem in this forum.
Instead, Google Nest either disables the "Next" button or displays a message saying the invitation can't be sent right now.