no, the HSA can be in either name. as a matter of fact if it's in your name originally and you transfer it to an a/c in his name you had an impermissible distribution subject to taxation.
if either spouse has family coverage for a month both are treated as having family coverage for that month. each HSA account can have only one owner. there is no such thing as a joint HSA account.
so if there was family coverage all year the maximum contribution is allowed and can be split any way you want between your account and his.
the question about coverage refers to the coverage of the health plan that the owner of the HSA had for the month. your a/c can be used to pay his medical expenses and vice versa.
