When you say taken you aren't selecting a company name based on what's available as a .Co.uk are you?
I do this for a living, branding and naming workshops, so please ignore the domains.
You can add thexxx or underscores etc there is always a way around the domain, and most users now click straight from links rarely does anyone type in a full domain.
If she is going limited then she will have to register a business name and again you can trade with a different name.
So start with what suits, names, grandparents names, family words something that works for her and is a bit personal with a story.
We had a cleaning business called Gladys after her granny.
And always pretend to answer the phone and stay the name out loud.
Also it will most likely be the first name you thought of or the one you are using as a temporary name!
Good luck