HSA accounts are individually owned, so in the name of that one person. There is no "family HSA" account ownership provision. The account owner has to be the insured by an HSA-qualifying plan to be able to open that account. The Amount to put into the account is driven by if the coverage is individual or family. Once the plan changes or no longer applies, the HSA stands as fully functional but cannot accept new contributions.

Medical costs can be paid from that one account on behalf of any person that would be considered a qualified expense.