Female mosquitos (male ones don't bite) inject their saliva in order to get as much blood out as quickly as possible. The saliva contains an anticoagulant to prevent clotting. Most people are allergic to some proteins in mosquito saliva, which triggers the immune system to send in histamines. One of the things those histamines do is to dilate blood vessels, allowing for white blood cells to get to the area more quickly. Dilated blood vessels help the mosquito to get more blood.
So there might be a benefit to mosquitos from their bites causing that itching, because of the dilated blood vessels that come first.