Friday first appeared in Iron Man Vol 3 #53. Friday, who was named as a reference to a Friday Girl, was created as a back AI for JARVIS. However, she had the attitude of a teenage girl and rebelled against Tony in a "You're not my real dad"-sort of way. After Tony grounds her she does eventually cool off and returned to her duties as Tony's assistant.
Friday, renamed as F.R.I.D.A.Y., appears once again as a backup AI for JARVIS in Avengers: Age of Ultron, with the only main difference being that she trades in her young age for an Irish accent.
*Thanks go to TylerMirage for the Jennifer Hale clip*