I don't remember the place I found this. Okay, my name is Abby, NOT Abigail. Says so on my birth certificate and liscense.
Now, Abby does not mean "father's joy" or "source of joy" or any kind of joy for that matter.
Allow me to elaborate.
Abby/Abbie/Abbey
-I shudder at the name Abbey, makes me think of a building.
Abby is derived from Abigail
Abigail is derived from Avigayil, a hebrew name
Av i gayil
Av - father
i - "of"
gayil - joy
So, you see, Abby, obviously hailing from ABIgail, and AVIgayil, means "father of". It is the gail/gayil part that means joy.