My understanding is that the user you are trying to add must be signed into iCloud and have keychain turned on in the iCloud setting. (I think that setting is off by default.) There might also have to be some sharing settings set to share.
Supposedly, people who meet these criteria (and others criteria?) are supposed to show up black in the list when you begin to enter. If they do not meet the criteria their entry is yellow. Supposedly, previously designated family members are supposed to show up in the list even before you start typing.
I had problems adding a family member and resolved it like this: I deleted the failed invitation. On family member's phone I "removed" their default home that was created by their own phone. (I don't mean the home app, I mean delete the "home" within the app that was set up and showing home pods.) Then, I resent the invitation and it magically appeared as a banner notification. I accepted it and voila - they are now able to access the "home" I set up with all its devices.