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The main character, a girl, and (I think) her mum are on the run for something and they move around to different cites. Every time they enter a new city the main character goes to a library, closes her eyes and runs her finger along the spines of books on one shelf. Her mum tells her to stop and she picks the book that her finger is on, and assumes the name of the first female character. For most of the book her name starts with C. It's also a weird name so the main character is extremely unwilling to use it.

When did you read this? Where did you find it? How did you find it? Do you remember anything about the cover? Was it mystery, suspense, horror, coming of age? Where does the story take place?

My first thought was What Happened to Goodbye BUT, she lives with her dad, estranged from mom, but is always changing her name and becoming a "new person" with every move (and they move a lot), so that's not it.

And in the new town she gets a really old fashioned name like Chasity or something?

Fake ID

Dora has left the group but I agree this is "Fake ID" based on GR review excerpts:

Chastity Pureheart is a girl without a past. Everything about her is a fabrication, including her name. Along with her mother, Chass has spent as long as she can remember moving from one place to another with only a moments notice. She has her routine down pat - suitcase packed and ready to go at the front door at all times, hop in the quiet car and drive to where her mother has pointed to with her eyes closed on the map, stop at a library to draw a book of the shelves and choose a new name.

and

Every town she has moved to, she changes her name.