What is the meaning of "I read a book once"

Solution 1:

The writer of the "missed connection" note is grasping at straws (<- useful English expression) to create some kind of connection to the person whom he encountered on the L train. He knows nothing about her other than that she was reading a book and had some bruises. "I read a book once" is a clumsy, funny way of saying, "Hey, I have something in common with you! You obviously read books, because I saw you reading. And you know what? Some time in my past, amazingly, I also read a book! (So that's enough for you to contact me and maybe we can go on a date.)"

Solution 2:

It’s saying the speaker has only read one book ever.