[chumbucket]

You're on the right track. It's most likely the solenoid. You didn't mention what brand outdrive you have but I'm assuming it's a Mercruiser. Most Merc's utilized a Prestolite trim pump system and have two solenoids. One for the raise circuit and one for the lowering circuit. There are ways of checking with a multimeter but the simplest method is to short circuit the solenoid. the two large posts are where the current passes through to energize the coil which will allow for the voltage to flow to the raising (or lowering) circuit. So, the easiest test is to jump the two posts and it should allow the outdrive to raise. the solenoids are a very common item and not very expensive (thankfully unlike everything else with boats). The Prestolite motors and pumps are very reliable and last a long time.