There doesn't seem to be any advantage to the physical reaction of itching to a bug bite -- not the tickly itchiness of a bug that's wandering around on your skin right now that makes you want to scratch or swat it away, but the itch that appears hours or days later. The bug has already taken its fill of blood and gone away, so there's no advantage to the bug if we spread its saliva and anticoagulants around, and scratching doesn't seem to help heal the bite any faster -- if anything, it makes it worse.

My guess is that maybe itching is unpleasant enough to motivate us to try to avoid bug bites (and thus diseases) when we can, but not debilitating, so that we're still able to try and run away from more likely lethal threats like a bear coming at us.