It is a multi-step process with no guarantee of success, but my experience has been encouraging. First, examine the board carefully and identify where the goo is present. Second, desolder all the old electrolytic capacitors. Third thoroughly clean the board. Various methods have been suggested/recommended. Personally, I like to use flux remover and a nylon toothbrush, followed by 91% isopropyl alcohol and a toothbrush followed by a distilled water rinse. However, the dishwasher method seems to work pretty well.
Fourth, replace the capacitors with new ones.
Fifth, examine the board carefully and probably under magnification. Some kind of poking tool such as a dental pick is helpful and a continuity meter (ohmmeter) is pretty much a necessity. You are looking for areas where the goo from the capacitors corroded traces and vias on the circuit board and pins and solder on the various components. Concentrate your examination in the areas where the goo was found and where the malfunctioning circuity is. You'll probably find some devoured bits which can be bypassed with a bit of wire wrap.
Back around '95 my IIci just quit booting up. The above plus bypassing a dissolved via with wire fixed it at a time when a IICI was worth close to $1000.