Unlike physical strength, emotional strength is very subjective in my opinion. To me, the emotionally strongest people are the people that deal with extreme emotional pressure, non-physical abuse, mental illnesses etc, and still come out on top. I have a friend that has schizophrenia and severe clinical depression and she's one of the strongest person I know because she's able to live everyday and survive her emotional constraints. Yes, there are people who are much more stable emotionally but they have never been placed in a position where they really had to fight for that stability, which to me doesn't make them stronger than someone else just because they "deal" better.
That being said, I think in BtVS, Buffy is the one that is emotionally the strongest. I mean, she's gone through isolation because she's the slayer, she's had to kill and see her loved ones die, she went through clinical depression, her destiny was dumped on her and she had to face death at 16, and she's faced abusive relationships. It's a miracle that this girl is still standing. She might not always be the most stable of the gang, but she's by far the one that has dealt with the most by s7 and she still comes out on top.
For AtS, I'm going to have to go with my man Wes. Very similar reason to Buffy. The guy lost everyone that was ever close to him and pretty much had a super shitty life up to the moment of his death. In second I would place Fred because the girl was literally shipped in some crazy dimension, you need to be dam strong to survive that and still be sane.