Assume ZF. Consider the claim:
(1) For any infinite set $\Omega$, there is a finitely additive probability measure $\mu:2^\Omega\to[0,1]$ with $\mu(A) = 0$ whenever $|A|<|\Omega|$.
Then (1) is implied by Hahn-Banach and the claim that the union of two sets of cardinality less than $|\Omega|$ has cardinality less than $|\Omega|$ when $\Omega$ is infinite. (Let $\scr B$ be the boolean algebra $2^\Omega$ and let $\scr I$ be the ideal of all subsets of $\Omega$ of smaller cardinality. HB is equivalent to the claim that every boolean algebra has a f.a. probability measure, so $\scr B/I$ has a f.a. probability measure, which lifts to a f.a. probability measure on $\scr B$ with the requisite property.)
[Deleted Question 1 as it was predicated on something false.]
Question 2: Is anything known about the strength of (1)?
A related claim is:
(2) Every set has a f.a. probability measure on its powerset which is not supported on any set of smaller cardinality (i.e., if $|A|<|\Omega|$ then $P(A)<1$).
Clearly (1) implies (2) (at least in the infinite case, but the finite case of (2) is trivial).
Question 3: Does (2) imply (1)?
Updates: The answers below so far show: (1) is equivalent to AC. ZF+DC is not strong enough to show (2). It's still open in this discussion whether (2) is equivalent to AC (I doubt it).