You seem to be a bit undecided. On the one hand, wanting her to go, but on the other wanting her to stay and pay rent, which is understandable if your gf's work is freelance and that's dried up and receiving a bit of rent would help her out of a financial hole.
Your gf needs to decide what she wants N to do, though, whether she wants this person (clearly not a friend, they're a piss-taker), to stay and pay rent, or whether she wants her to leave.
But I suspect that while formalising the situation in a way by asking for weekly rent would help your gf with her finances, she'd still be resentful of the impact on her and her son's home life.
Your gf is effectively subsidising her so-called friend's holidays and visits to family. I had a similar situation, let a friend stay rent free in a spare room as he didn't have a job to start with, and then he got a job and then he booked a holiday. In hindsight, I ended up thinking, Hang on, I can't afford a holiday, I'm skint... and then it struck me that in a way I was subsidising/helping to pay for my friend's holiday, because they were able to go on holiday because they weren't paying me rent. So they had money in their pocket that they otherwise wouldn't have had because I was helping them out/effectively subsidising their lifestyle.
On the one hand, it's understandable to be sympathetic towards someone who's only working part-time. But given that she's not paying any rent or bills, she's effectively as well-off as someone who's working full-time. Someone working full-time might be paying around half their income in rent and bills in London, and so their disposable income on a full-time job might be roughly equivalent to the freeloading friend's disposable income from only her part-time job.
You said that this friend has cooked an occasional meal, but has she been contributing to other household costs, like buying loo rolls and washing up liquid and clothes washing detergent and tea and coffee and milk and stuff? Or has your gf also been subsidising N's lifestyle in that way too? Maybe it's not just that N isn't contributing, but maybe she's also leeching off your gf too and adding to your gf's living expenses too? And while any one of those things might seem like a minimal thing, but when it's lots of little things over months and months, it all adds up.