Theory!:
I feel like Finne is an illegitimate child of a noble. Like maybe her mother worked in some noble's manor & there had some kind of affair with a noble there. This could also explain as why there were assassins after her & her mother, because if left alive Finne could have easily been used as a blackmailing device by her mother to get excepted into the noble house. Or the fact that a noble had an affair with a commoner if reveled could bring shame to the noble household.
The reverse of this situation could also be the case, and that Finne's mother was a daugther of some noble who fell for a commoner & had a child with him. Maybe her family didn't like the fact she got pregnant with a commoner & so (for various reasons) she decided to run away. This explains how she knew proper etiquette despite of being a commoner and why she was soo willing to teach them to Finne. Perhaps the assassins were sent to kill Finne (which could bring disgrace to her mother's household in the high society due to being a low born) & retrieve her mother to perhaps get married of. I guess it's also possible that the assassins were sent to kill both of them (the mother & Finne) as keeping them alive could have been troublesome.
(Both of these theories make more sense because Finne has the ability to use magic which only nobles are supposed to have. Plus with how the story is being set up the main problem between Bard & Finne's relationship is probably going to be their difference in status & wouldn't it be really convinient if Finne turned out to be a child of some noble??)