Her cleaning the house every day is not necessary.
So I was wondering, why use her instead of she in that gerund phrase? Is that actually correct? And can I say "The maiden cleaning the house everyday is not necessary" instead of using her?
Her (constantly) cleaning the house every day is not necessary.
The expression "her cleaning the house every day" is a non-finite clause with a personal pronoun as subject.
Non-finite clauses take an accusative subject pronoun like "her", not a nominative one like "she".
Note that "cleaning" is a verb here, not a noun, as evident from (a) the possibility of it being modified by an adverb like "constantly", but not the adjective form "constant", and (b) the fact that it has a direct object, i.e. "the house".
This is an interesting one.
The traditional answer is that her is the correct choice - but it is a possessive her, not an objective one. So The maiden's cleaning the house is the corresponding form.
However, many people don't use the possessive, or only use it in formal speech, so Her cleaning and The maiden cleaning would be normal for them.
What nobody says (or almost nobody) is She cleaning the house every day.
(People would say She, cleaning the house, found something unexpected, where "cleaning the house" is a participial clause with adverbial function, but that is a different structure.)
I think the reason that nobody would say "she" is that, however hard grammarians have tried to pretend that Modern English has case inflection on pronouns, it doesn't, except in an artificial "learned" variety.
Joseph Emonds argued in his 1986 paper Grammatically Deviant Prestige Constructions that the choice between "subject" and "object" pronouns is structural, not case-based, and this is one kind of example that tends to confirm this: even if some of us have internalised the traditional case distinction, and naturally say he and I will go but between him and me, we still don't use I or she in sentences like your example.