We stayed at Giuseppina's with our little daughter (2y). It was perfectly located - quite close to parks and not right in the city center (although next to busy street) st. It is a modern and renovated apartment, nice design and not too many furniture. Although we arrived late, Giuseppina was waiting for us, provided all necessary information, directions and nice breakfast incl local fruits.
AS we stayed 10 days, the washing machine was missing a lot and there are no public laundries in Palermo. Also, the kitchen is not heated, but i guess this is not a problem in the summer. Be aware of the electricity which goes off if you plug in too many things at the same time.