We'd need GOOD pics of the board to see what components are on it.
"Most" don't mind getting wet, but some do.
"Some" of those that mind, can be covered.
"IF" you can't get it wet then you need to use electrical contact cleaner after the initial prep steps of getting the majority of the big particles off.
"IF" it has an EPROM you need to be careful about stray ESD using a brish.
"IF NOT", you can use a dry brush to first get off the bulk of the material.
"IF" the board can get wet, you can use hot, strong, detergent solution and submerge the board and use a soft bristled (paint, etc) brush to dislodge gunk. If it's can't get wet, you'll be doing that with residue-free electrical contact cleaner.
"IF" the grime is only cosmetic except for the the electrical contacts to wiring harness, you may not even need to get the whole board clean, just those electrical contacts. How to clean those depends on whether they are gold or silver plated and just need grime cleaned off, or tin with a median amount of oxidation, or brass with a lot of it. Tin can usually be cleaned with a paper towel/etc and contact cleaner. Brass needs a brass cleaner then a rinse if it leaves behind an anti-oxidative film, or abrasion cleaning if it is more readily exposed to facilitate that.
Relays, if not sealed, should not be allowed to get contact cleaner inside unless it is a last ditch effort because they are shorting out and you need to use the equipment prior to having a chance to replace the relay. Relays that are sealed but have a breather hole covered with tape, might have the tap fall off if sprayed with contact cleaner. Worse things can happen but it is best not to get any inside.
If it's a full blown mainboard with a computer real time clock, the battery should be removed prior to exposure to water. Any socketed chips should be removed, following ESD safe procedures, before cleaning then reinserted after the board is dry.
After a detergent wash you can rinse with water that has alcohol in it to break the surface tension, or a drop of dishwashing machine rinse agent to break the tension, or worse case shaken gently, then blown dry. If it is a modern construction using water soluble flux, do not let pools of dissolved flux sit in any area while it dries as they can cause corrosion and short circuits.
If it has rosin core flux residue left behind and you use alcohol, then you should similarly make sure there are no pools of that muck left behind unless it is very hard from old age, not flowing anywhere.
Be very careful with compressed air, that velocity is not really needed and if the board flexes too much you risk breaking solder joints or components, depending on what's on it.
The next consideration is why it got so filthy. Does it have passive heatsinks and needs airflow to stay cool enough or was it just a substandard job sealing it from the elements? You might see whether you can improve that.
That's about it, I shouldn't have written this much before seeing pics.