I was playing with my dad and he said that he didn't think that Maester Aemon could be used to save himself. I know that he is wrong, but there are no rules on the FAQ or posts here because there is nothing controversial about it so he won't back down. I need help convincing him that Maester Aemon can be used to save himself. Thanks!

Tell him to read the card literally.

"If it is not Winter, kneel Maester Aemon to save a Night's Watch character from being killed. "

Is it not Winter?

Can Maester Aemon be knelt (is he standing)?

Is Maester Aemon a Night's Watch character?

Is Maester Aemon going to be killed?

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If these are true, then you can kneel Maester Aemon to save himself!

Or, attack it another way. Point to something like the "Stewards" agenda (Night's Watch characters you control gain an INT icon.) or the event "Yoren's Task" (Any Phase: Discard 1 Night's Watch character from play to search your deck for 1 different Night's Watch character,reveal it,and add it to your hand.). Ask him if those cards would consider Maester Aemon to be a Night's Watch character. When he says "yes," ask him why effects like that look at Maester Aemon's trait and count him as a NW character, but his own text (Response: Kneel Maester Aemon to save a Night's Watch character from being killed.) would NOT look at his trait and count him as a NW character.

You are correct that there is not FAQ entry or rule on this because it is so basic. When an effect refers to a trait, all it looks at is the trait. It doesn't matter if the effect and the trait are on the same card. This happens all the time. Melisandre (While Melisandre is standing, Asshai characters get +1 STR.) gives herself +1 STR while she is standing because she is an Asshai. Bringing Varys (After Varys comes out of Shadows, choose and discard 1 Ally character from play.) out of Shadows when there is no Ally character in play is a bad idea because, as he has the Ally trait, he will be forced to discard himself.

Essentially, have your father point to the rule that says characters cannot work on themselves. He's going to have a hard time doing so, given that there are so many that do.