Jarvis was a very sophisticated artificial intelligence but essentially was software, right? It would seem uncharactistic (and kind of dumb) if Tony not have multiple backups of Jarvis. So why did he still replace him with a different AI after putting him into the Vision in Avengers: Age of Ultron?
Because Tony is a complicated person.
A bit of backstory. MCU J.A.R.V.I.S. is a backronym for Just A Rather Very Intelligent System. It's a backronym because the real Jarvis was Howard Stark's butler. He's seen in Agent Carter, and in the Iron Man 2 tie-in comics. Jarvis has been a real person and an AI butler for Tony in the main 616 and ultimate 1610 comics. Tony names J.A.R.V.I.S. after the deceased Edwin Jarvis, as a tribute (in the tie-in comics above).
So, when Ultron tries to kill J.A.R.V.I.S. during Age of Ultron, Tony feels a personal loss (as do the others for some reason). When J.A.R.V.I.S. comes back from the internet at the Nexus, and uploads into the Vision body, Tony's attachment to J.A.R.V.I.S. as a individual simply prevents him from pulling a backup from a shelf. Would you clone your best friend for selfish reasons? Well, Tony wouldn't. It's not like J.A.R.V.I.S. is dead.
So the choice to use F.R.I.D.A.Y. (The acronym has not been disclosed yet) is a touch of personal respect towards Jarvis, the AI and the person.
Also, Remember that Tony, coming out of Iron Man 3, and as explained in Civil War, is still suffering constant issues from his habit of building Iron Man Suits, PTSD, stress, etc. He changes his looks some, he has to build a new house, he initiates Clean Slate protocol and destroys 52 odd suits, is on the outs with Pepper, etc. Not going with a J.A.R.V.I.S. backup is just another sign of his mental issues.
Out of universe:
In the comics and various cartoons, Tony has gone through many Artificial assistants. J.A.R.V.I.S. is just the latest (he was pulled from the movies into the main comics). Others include H.O.M.E.R., P.L.A.T.O., V.I.R.G.I.L., and JOCASTA (basically female Ultron). Most importantly, FRIDAY was around for a while as well. So this is also the movies giving a nod to comic continuity.
From Age of Ultron, Tony went with FRIDAY over Jocasta and most likely Tadashi (a reference to the other Marvel comic and movie, Big Hero 6, where Tadashi created Baymax). Being Tony, a hyperactive personality, he's likely writing new A.I. all the time, as often as he makes Iron Man Suits or any other new technology.
The more practical out of universe reason is that, as others have mentioned, having the same actor for two characters would be confusing, or expensive. From an interview:
During an interview for "Age of Ultron" with Business Insider, Bettany said if it wasn't for director Joss Whedon, we very well may have seen another actor play Vision on the big screen.
"They [Marvel] have a rule, that you’re not allowed to play more than one character in the Marvel universe," explains Bettany. "So, they broke the rules … they bent the rules slightly for me because Joss really wanted to bring the Vision into the film world, and he really wanted me to do it."
The 43-year-old actor described the transition from A.I. to a physical presence in the films as a bit of a double-edged sword. "Before I used to turn up for two hours and get paid a silly amount of money for … I used to feel like a pirate running off with a bag of cash at the end of the day," jokes Bettany. "Now, I actually have to show up and stuff."
So FRIDAY was chosen because out of universe the role was retired so the voice actor could become the actor for a different role. The in-universe action was shoe-horned in without an explicit explanation of why J.A.R.V.I.S. wasn't backed up.
You sorta answered your own question. FRIDAY is a backup for, and probably at least partially of JARVIS. It would be kinda weird having the same voice for both JARVIS and Vision. Not to mention FRIDAY has been improved to be fully autonomous. Vision is fully autonomous because of the Mind Stone present in it.