I can recommend a stay at Luca's apartments. The apartments are located at the edge of Lipari's center which makes it nice as you are not directly within the center of action, but still close enough by the center (city center & port 5-15 min walk) to have several supermarkets, restaurants and bakeries within walking distance.
The apartment facility holds a few apartments and is very neat! with a lot of lemon trees making you feel right away relaxed. The pool holds very clean sea water and is fantastic for relaxing, also here you can find lemon trees surrounding you. The apartment itself is furnished in a typical italian style, and very clean! Luca is a great host, whom you can call anytime. He is happy to help and will give you any recommendation you need, he also speaks English quite well.