There are many myths surrounding the problem of mosquito bites, especially when it comes to curing their itch. Some widely-circulated ones include rubbing bites with a banana peel or soap, applying toothpaste, covering it with sticky tape, or ground-up aspirin. In order to figure out what might work, first we need to figure out why they itch.
Female mosquitoes require blood meals when producing eggs (Males feed on nectar instead of blood, so they don't bite). But they don't actually bite, they have a needle-like proboscis that jabs into your skin to draw blood out like a syringe. To keep the blood flowing while they suck, the mosquito injects a bit of its saliva that contains an anti-clotting chemical.
As a result, the itching isn't even caused by the proboscis, but by your own body's immune response to the mosquito saliva. Your immune system releases histamines to destroy the foreign substance, which causes swelling and itching. Fortunately, the itch only lasts a few hours, so your best bet is just to wait it out. But if you really can't stand it, doctors recommend ice to reduce swelling and topical anti-itch creams for the itch. If bite symptoms last a very long time or are more severe than normal, consult your doctor right away.