Cate named Lux.
"Lux" is exactly the kind of dopey name that a teenager would give a baby, and since Lux was never adopted by anyone it's hard to imagine who else would name her that. So even if Cate never considered keeping her, she still felt enough of a connection to name her. If Cate doesn't seem to recognise the name, it's because she realised how dumb it was and expected her to be renamed.
- Jossed by the second episode, sorry: a nurse named her Lux because she had blonde hair but came from two dark-haired parents (Lux meaning "light").
- Even though this was explained In-Show, perhaps the writers also came up with the name Lux because of the title |L|ife |U|n|eX|pected...