Queen Victoria had two first names, Alexandrina Victoria. By which name was she known, privately and publicly, prior to her ascension to the throne? Also was Victoria a well known name and, if not, was it considered to be an appropriate royal name at the time?

Victoria was called "Drina" as a baby, and she started being called "Victoria" at the age of five.

The story of Victoria's name is kind of amazing. Her name wasn't chosen for sure until her christening, partly because the Prince Regent (who became George IV) was pretty nasty to her family. It had been proposed that she be named Georgina Alexandrina Victoire Charlotte Augusta. Her mother had suggested Georgina, in honor of the Prince Regent. Her father wanted Victoire, as the French version of his wife's name, Victoria. Her mother also liked Alexandrina, for her godfather, Tsar Alexander of Russia. The compromise name Georgina Alexandrina Victoire Charlotte Augusta were sent to the Prince Regent with several days left to the christening for his approval... and he didn't respond.

When the day of the christening arrived, there was still no consensus. Finally, during the ceremony, the time came for the Archbishop of Canterbury to ask for the baby's name. It's usually a formality, but at this point, nobody actually knew what her name would be. There was silence in the chapel. Finally, the Prince Regent barked, "Alexandrina." He refused to have his own name be used. The baby's mother began to cry. Her father suggested Elizabeth, which was also shot down. At last, the Prince Regent relented and said they could also use the mother's name. Hence, Alexandrina Victoria.

According to the biographies I've read, Charlotte was struck because it was the name of the former heir to the Prince Regent, his only legitimate child, who died tragically in childbirth along with her baby (I can imagine he didn't particularly want the child of a family he didn't like to share a name with his dead daughter). Georgina wouldn't do either because the Prince Regent refused to have essentially his own name beside the name of the Tsar of Russia, or just because he didn't like the family.

The story has something like a happy ending. The Prince Regent, who was later George IV, was actually quite generous to little Victoria as she grew up. Knowing that her mother and John Conroy, her mother's comptroller (something like a high-class secretary and accountant), kept her under strict control, he would swoop in and take her away from time to time, and she fondly remembered a carriage ride through the city with him for years to come.

Edit: I realized I conflated George IV with William IV for a historical tidbit, so I removed that. Woops!