What do the letters ayin, bet, gimmel stand for between the groom's name and the bride's name on a Jewish, Hebrew wedding invitation? In the past, I've seen ayin, bet, lamed which I believe stands for "im bechirat libo" or "with the one chosen by his heart"
"Im Bat Gilo" -- very roughly, "with a woman suited to his nature."
The Gemara Nedarim 39b says that a hospital visit is especially efficacious for the sick fellow if the visitor is "ben gilo" with respect to the visitee. Rashi (or whatever medieval commentary there pretends to be Rashi) says simply -- "roughly the same age, not a young man visiting an old man or vice versa." Rosh, however, says the visitor needs to be born under the same astrological sign as the visitee; presumably otherwise, I guess the karma (or the personalities?) don't "click" the same way.
So let' just leave it loosely as "with the person suited to him." My impression is this was the more traditional wedding notation, at least printed in the direction from the groom to the bride.
"Im bechirat libo/liba", "with the one his/her heart has chosen", as you indicated, sounds easier to our modern ears; it's gender-neutral; and it avoids the spooky stuff; so many modern invitations simply put this in both directions.
A rav explained to me that the word "Gilo" come from "gil" related to "gilah" meaning "joy". So, "Im bat gilo" means "the daughter of his joy".
The concept seems slightly Kabbalistic, in the sense that one's true fullest joy is hidden until he gets married. When a man finds his Kallah, she draws out his inner joy, so that is the sense of being "bat gilo".