In The Legend of Korra 4x02, "Korra Alone", Korra meets Toph for the first time and they have this exchange:
Korra: A spirit led me here and told me I was supposed to find someone. Is that you?
Toph: Beats me. But if your Avatar senses called you here, you should probably listen to them.
Korra: Wait, you recognize me?
Toph: In a manner of speaking. We were good friends in your previous life.
Korra: I can't believe it ... Toph?
Toph: Nice to see you again, Twinkle Toes.
But how did Toph know that Korra was the Avatar?
She is blind and recognizes people through her "seismic sense" of using earthbending to feel vibrations in the earth, so had she met Korra in the past (e.g. when she was a child), it's likely she would have recognized her by the sort of vibrations that she gave off. However, Toph's dialog indicates that she recognizes Korra because she knew the last Avatar (Aang), rather than from them meeting in the past.
The issue is that Korra is the reincarnation of Aang (i.e. same spirit, completely new body), and I'm pretty sure that one's reincarnated spirit wouldn't give off vibrations that Toph could recognize because it's not a physical thing. Furthermore, the two Avatars have very different personalities, so it's not likely that Toph could have recognized Korra by that.
So how did Toph know that Korra was the Avatar?