Just going by memory of usage pattern:

Gadget: a small (at least usually small enough to be carried around and utilized by a single person) piece of technology for a specific, often niche use. Eg, "go go gadget shark-repelant spray gun!"

Contraption: a device, often jury rigged or prototyped, built to serve some particular purpose. Eg, "I built this here contraption to dig us a way out of here"

Doohickey: A word used refer to a contraption or other piece of technology that the speaker does not understand, or does not know the purpose of, especially to emphasise that lack of understanding. Eg, "they're gonna 'beam' us out of here with that doohickey."

Not sure about gizmos.

Edit: examples

Why are we not talking about thingamabobs here?

I may be wrong but I think each of their descriptions scale differently depending on how much skookum they have.

Honestly, most people will use those words interchangeably. So if you're talking to someone, just understand that, to that person, they likely mean the same thing.

Also, what about a gewgaw (or is it a geegaw)?

Any one of those phrases could be used to describe The Device

Gizmo is no hydrohomie, that's all i know.

A Gizmo is also a Gadget, but with a forgotten or unknown name.

syn·o·nym /ˈsinəˌnim/ noun noun: synonym; plural noun: synonyms

a word or phrase that means exactly or nearly the same as another word or phrase in the same language, for example shut is a synonym of close.