Originally a script was written based on the characters and story of the first game:
Romero's screenplay revolved around the plot of the Arklay incident and included characters from the Resident Evil video games. Chris Redfield and Jill Valentine were the lead characters, involved in a romantic relationship. Barry Burton, Rebecca Chambers, Ada Wong and Albert Wesker were to also appear.
Romero's script was disapproved of and production was placed into development hell.
Then Sony decided to hire a different writer:
Paul W. S. Anderson wrote a screenplay, which was ultimately favored over Romero's. In late 2000, Anderson was announced as director and writer, and Resident Evil re-entered pre-production stages. Anderson stated the film would not include any tie-ins with the video game series as "under-performing movie tie-ins are too common and Resident Evil, of all games, deserved a good celluloid representation".
So the character of Alice was created. Paul W.S. is said to have based Alice on the strong women which are portrayed in the Resident Evil game series. Because, as noted in the original question, the film was intended to originally be an allegory to Alice's Adventures in Wonderland, the character was given the name Alice. Although that name is never spoken in the first film, it is the name listed in the credits.