Option A is saying that the aunty has been visiting for three weeks and is still at the house. The word 'has' is present tense, so it is saying that she is staying at that time. If you said 'My aunt has NOT been here for three weeks' then it would be the same as the first statement 'It's been three weeks since my aunty stayed here.'

Even though I'm not a native speaker, I lived quite some time in UK and B is definitely the right answer. Preterite is used when an action is done or should've happened but didn't. Here, the sentence says "three weeks SINCE my aunt STAYED here", meaning the narrator's aunt left 3 weeks ago and didn't come back.