I usually have a way with nicknames, but they're often based on what someone does/has done or typical behavior. Just playing with the thought, there are a few things that could work for the name Brooklyn. I'm not sure if Olivia as a character would use them, but anyway...

"Lynnie" as a play on "lyn", but it's quite impersonal considering there are many names that end with "lyn". Another one is "Boo" as a play on Brooke. However, it would be more suited for a boyfriend of Brooklyn's. You know, "Boo" as in 'an affectionate/significant other'.

Olivia will probably just say Brooke! Plain and simple, I like it.

Take your time, no rush.