Okay, so we all know that River Song’s real name is Melody Pond. And, that the only reason why she went on to call herself River Song is because in Let’s Kill Hitler the Doctor reveals that that is what he calls her.
The Doctor says the name you choose is like a promise.
And I think that day because of whatever he whispered in her ear as he was dying, she chose that name and made her promise.
The promise to be his River Song.