I had a really beautiful stay at Arianna's place. The apartment 'Demetra' was quite spacious, everything was perfectly clean. The place was also beautifully furnished and cozy. It is right in the center of Lipari so you can either explore the town or enjoy the Island (I did so by renting a bike).
Arianna and her family were extremely helpful and always available.
Only con is that that the apartment is poorly insulated, so it was a bit cold at night (in March).
I can really recommend you to stay with Arianna!