When Lorna Bucket embroidered Melody's name on the prayer leaf in the language of the Gamma Forest, she used the closest translation she could. When the TARDIS translated the words back into English for Amy & Rory, "Melody" became "Song" & "Pond" became "River". (The language of the Gamma Forest has only one word for a body of water, because "the only water in the forest is the river".) The given name & surname were on opposite sides of the prayer leaf, with nothing to indicate which was which, so they were read in reverse -- as "River Song". (See "A Good Man Goes to War".)
Because of the circumstances of her conception -- in the TARDIS while travelling through the spacetime vortex -- she had some Time Lord characteristics. (The Time Lords originally got those characteristics in the same way, by exposure to the vortex.) Kovarian & company then enhanced Melody's Time Lord characteristics while she was still in the womb. As a result, Melody could regenerate. (Also "A Good Man Goes to War".) We saw her do so twice: once at the end of "Day of the Moon" & once in "Let's Kill Hitler", when she became the incarnation we already knew as River Song.
Melody/River is a time traveller & her mother Amy first met her before Melody/River had been conceived. (That kind of thing has happened before. Ace's mother Audrey was a little baby when she first met the teenage Ace in "The Curse of Fenric".)