Response: After you play Ygritte from your hand, choose and take control of a neutral or Night's Watch character until Ygritte leaves play

What happens if I take control of a Baratheon character who has the Night Watch trait but loses that trait when I take control (i.e. my opponent is running Bara / Winter / Wall)

When Ygritte leaves play and they ask me for their Night Watch character back, I could argue that I do not have any and therefor cannot give it back

The duration has nothing to do with the Trait . Otherwise, your opponent could argue that, since the character is no longer Night's Watch , you should give it back immediately. The ability means "[...] choose a Neutral or Night's Watch character. Take control of that character until [...]"

Said another way, the "neutral or Night's Watch" restriction in her ability is a targeting requirement. Once the target is chosen, it doesn't matter what traits or House affiliations the character gains/loses.

Wow, I never thought about how they worded the card. I can see why there's confusion.

They really should've worded it

Response: After Ygritte enters play, choose a neutral or Night's Watch character. Take control of the chosen character until Ygritte leaves play.

This works better since "the chosen character" doesn't change even if the traits or affiliations do.