Help! Character Refuses To Be Named!

So I need some help I’ve been toying with a story for a while and one character has refused to be named, adamantly (which is quite annoying) everyone else in the story is happily named (for the most part)
The premise of my story is a family, six adult siblings, all living their somewhat complicated lives and coming back together for their father’s birthday and discovering that he wants to sell the family house and move into a near by retirement village which causes the eldest daughter to buy the house off her father with the intent to fix it and have it stay in the family… she is the one that refuses to be named

Father - James ‘Jimmy’
Mother - Pamela
Step Mother - Anita ‘Annie’
Older Brother: James Jr JJ
Unnamed Character
Younger Brother: Andrew ‘Andy/Drew’
Sister: Lavinia ‘Vee/Vinnie’
Brother: Cameron ‘Cam’
Brother: Lucas (though open to suggestions/change) ‘Luke’

Unnamed Character: has been [name]Caroline[/name] - just didn’t click, [name]Camille[/name] - for a while this worked as her brother [name]Cameron[/name] was named [name]Paul[/name] but then [name]Paul[/name] refused to be [name]Paul[/name] and became [name]Cameron[/name] instead so [name]Camille[/name] and her brother [name]Cameron[/name] were so not going to work out for me, also tried on [name]Valerie[/name] (didn’t fit) [name]Elizabeth[/name] (too common and not right)
I really don’t know I want it to be appropriate for when she was born given that she’s about 38 in my story, someting not too out there but not super common, with a nice nickname that fits in with her siblings I don’t want an overlapping initial (which is also why I’m open to suggestions for [name]Lucas[/name]’ name as it overlaps with [name]Lavinia[/name]
ugh so hard because until I get a handle on her name I’m finding it hard to move forward

any suggestions?

EDIT: i tried to make it simpler to understand

I had to draw a family tree of this in order to get a grasp of everything.

Unnamed Daughter: [name]Veronica[/name], [name]Natalie[/name], [name]Natasha[/name], [name]Kara[/name]? I took the 1973 Social Security list and browsed its 150-300s. Maybe this is an idea?
[name]Lucas[/name]: Somehow this feels to classy and genuinely Greek to fit in with the siblings’ names ([name]Luke[/name] would be better), but that doesn’t help to resolve the double L problem. [name]Dylan[/name] (way down back then), [name]Eric[/name], [name]Sean[/name], [name]Nicholas[/name]?

Thanks I think I will look at the Social Security lists, I never even thought to do that
I tried to make it more streamlined and simpler to understand

What is a social security list?