The AnswerThis was a tricky quiz question after all, mainly thanks to the ITTF, who have just recently changed the rules dealing with this situation. I did warn you!
The correct answer to this question is the second option, where the umpire should do nothing, since it is legal for the ball to accidentally hit the finger and then the rubber of Player A. (Of course I am talking about a finger on the playing hand, not the free hand!). Under the old rules, the first option was probably the best answer - where the umpire should do nothing unless he is sure the ball hit the finger first and then the rubber, in which case the point would go to Player B.
The relevant and reworded ITTF Law is Law 18.104.22.168:
2.10 A Point
2.10.1 Unless the rally is a let, a player shall score a point
22.214.171.124 if an opponent deliberately strikes the ball twice in succession;
Notice the change - the ITTF have reworded the rule to make it clear that only deliberate double strikes lose the point.
I'm all in favor of this rule change - this will make everybody's life easier. Players can simply continue play when the ball hits their finger then the racket, and umpires will only have to look out for the usual deliberate double hits (which are exceedingly rare), any normal double strike that occurs when a player is swinging at the ball is almost certainly accidental. In the past, I've seen a lot of heated debates about whether a ball hit the finger and racket separately or simultaneously, and since it's virtually impossible to tell anyway in most circumstances, all we ended up with was a lot of upset players arguing over something that nobody was really sure about in the first place.
I can foresee there being a period of confusion and argument as players slowly become aware of the new rule change, so it might be worth while carrying a copy of the latest rules in your table tennis bag! And really, I think the ITTF could have done a lot more to publicize the latest rule changes - these are the defining laws of how the game is played - any rule change is a big deal and should be front and center on the ITTF web site.