The Amiga's IC's (the black thingies with many legs) that are inserted into sockets can be re-seated.
This means that you pull them up and insert them again, you don't need to pull them all the way out of the socket, a few millimeters are enough. This reassures a good connection with the socket again. The disconnection can be due to oxidation. If you want you can take them out of the socket, and give them a good cleaning with a rubber (as in pencil) and after that Isopropylic alcohol using i.e. a cottonstick.
It could be that you can't replace components yourself but read on as the following sections are very important. Get someone with soldering skills to help you if required.
You should examine the capacitors, those are the tall round ones shown in the image below. If they bulge on top, they need replacing.
SMD capacitors don't last as long as the through-hole electrolytes shown above. Generally they should be replaced now if they haven't been already. You can find SMD capacitors in A600/A1200, CD32 and A4000 computers. And they look like this:
Notice the solder pads in the above image, they aren't shiny but rather matte. This is a sign of the capacitor has leaked capacitor acid. This acid can damage the pcb and nearby components. It should be replaced as fast as possible to prevent further damage. Also a good neutralization using water and cottonstick, and/or isopropylic alcohol.
A2000, A500+, A500 ramexpansions, A3000 and A4000 have batteries mounted. These batteries leak as well as the SMD capacitors mentioned above, but they do far worse damage, Batteries should be removed at once. They can be desoldered or cut. A neutralization of the acid damage is neccessary! For this you should use either vinegar or lemon juice -the "acid" in these batteries are actually base, so a gentle acid should be used-. Clean after wards with water and then iso propylic alcohol.