In Asia they lubricate tiger balm or Chinese oil on the mosquito bite. I do always, itching disappears very quickly.
Do not have vinegar at home?
After sun is also an option!
Yes it sounds weird but it works. It really cools. Preferably with an after sun there in as many natural products as possible.
Stung by a mosquito? Use your nails or another sharp object to make a cross on the bite and the pain is gone! Really helped me!