Response: After you win a challenge in which Arya Stark attacked alone, each opponent with a Title that opposes your Title must also satisfy the Claim of that challenge, if able.
If you win a [MIL] challenge in which attached character attacked alone, instead of the normal claim effects choose and kill 1 character controlled by the losing opponent.
Just my two cents on the topic and I will preface this entire statement with… here is what I believe/think/guess/estimate:
There is nothing in Arya's text that makes the other "opposing" players "losers" of a challenge, though they are satisfying claim of that challenge. To me it's similar to a Red Vengeance that picks a 3rd person at a melee table (provided player 1 is attacking player 2, who uses RV to redirect claim to player 3). They aren't losing this challenge, though they are paying claim for it. They do not get to trigger their own Taste of Blood (or similar) effects.
The attachment allows you to choose claim on your losing opponent. So instead of them getting to pick, you do. But there is only 1 losing opponent. This replacement effect would come into play before you get to triggering responses, so after winning the MIL challenge against your opponent, normal claim is replaced with you decided to kill someone controlled by that losing opponent. The key here is "losing". When you get to Arya's Response nothing has changed who lost this challenge so I don't see any conceivable way you could think the Courser could be used against your "opposing" opponents also.
The next important thing to note is that Arya's ability is NOT normal claim, so the Courser could not be used to replace that effect. Normal claim is defined by the game mechanics and can be replaced by cards in play such as the Courser, but her ability is not one of them (or it would say "instead of normal claim… yada yada yada"). That's my take on her ability at least…
Someone else mentioned getting to pick 3 targets on the losing opponent's board, and after writing the above I finally see where they were trying to go with that.
If Arya's ability does in fact issue "normal claim" to the opposing opponents, but the Courser replaces it with "kill a character of the losing opponent" then you should basically be able to redirect Arya's additional "claim" effects into additional targeted killed effects against the single losing opponent. However, I'm not sure I would consider her ability to be applying "normal claim" to your opposing opponents.
How does Red Vengeance work against claim replacement effects?
As an additional question, with or without the Courser, how many times does the loser of the challenge have to satisfy claim if his Title is opposed to mine and I trigger Arya's response?
I'm not sure I've seen a direct response to this, but if you attack an opponent you get normal claim, then you trigger Arya and any opposing opponents must also satisfy that claim. I would say the opponent that opposes you would have to satisfy the claim twice.