Pop went with Finley against the Thunder because they have a surplus of length at the wing positions, with Durant, Sefolosha and Green (though he's really a combo forward). Mason is really about 6-4 (listed at 6-5) and doesn't have the length to guard any of these types of players. Finley, while an even worse defender, is at least 6-7 (I don't mean at least in the sense that he might be taller than that, I mean it in the sense that at least he has that going for him), so as long as he can do a half respectable job at keeping those guys (minus Durant, who Finley should never ever be matched up against) in front of him, he has the length, when he contests, to at least make things more difficult on them than Mason.
This has been an ongoing problem for years now: the Spurs don't have anybody to guard these types of players. Rather than addressing it, they continue to throw anything they can think of against the wall and hope it sticks. McDyess, Bonner, Jefferson, Bogans, Finley, Hill, all generally get their chance against these types, but not one is suited to guarding them. Although, Jefferson, specifically against Green, shouldn't be a bad match-up.