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It must be a Chevy engine of one variety or another in 04

What did you do back then? It sounds like you were just happy that it started  and it's been good for 5 years or so I guess. 

I'd be checking all of your cable connections at the battery, starter and starter relay and the negative cable at the battery and the engine grounding point at least as a place to start.

You could use jumpers to put power direct to the starter, you could pull the starter out and use jumpers on the workbench or take it to an auto parts store to get tested.

There are a variety of ways to use the jumpers, but I'd check the connections first, then ask about jumper hookup.

The click is probably either the starter relay or the solenoid on the starter

A whack on the starter with a hammer can wake them up if that is the problem.   Hit the starter body not the solenoid.  Been around a few boats with the rusted starter syndrome.   The hammer trick does not fix anything but it can get it working again if it is just stuck from rust.

With the newer permanent magnet starters like you have, the whack can turn a good starter into a junk one if you hit it hard enough to break a magnet. A tender loving whack if you must  

Clean the connections, make sure they're tight like mentioned earlier as a starting point.

On an 04 the starter relay is just like the fuel pump relay and the ECM relay

Do the books that Dad mentioned previously have a wiring diagram you could follow your way through?

Since it had the same problem 6 years ago and is stored out of the weather, I bet Ken's suggestion on connections finds the problem.  Other than that, you say battery is charged, that doesn't mean the battery is good, so after checking connections you could jump from a good battery and see if it turns over.  Also if there is a battery switch check connections there as well.

Ken, what's this about gentle taps to the starter?  When in a Dunkin Donuts parking lot under my '98 GMC diesel pickup there was nothing gentle about my hits to the starter with a 15" adjustable wrench. And yes, it got it started, more than once.

Some people might even figure there's a problem and get a new one or get it fluffed and buffed/rebuilt after the second or third time you have to smack it instead of waiting for that cold, wet, winter day  

A small part of my brain remembers being that guy, but it was a WalMart parking lot

The older starters with electromagnetic field windings can handle a beating more than the newer permanent magnet style that have the magnets glued to the housing. The older ones have field windings bolted to the housing

Not sure what you have for a starter, but I'll guess it's the older style.

I imagine that we're probably keeping James entertained 

Just for laughs, same brother restored a 1978 Keaton Jet boat with his son.  They did a beautiful job, all new carpet, interior, gas tank, gauges, exterior was all re finished to look new.  As he did this job I advised him to change all the hoses and belt  on the engine.   

Jack Keaton who built all the Keaton Boats had a great following in Sacramento.  When he passed away there was a memorial on a Riverboat on the Sacramento River.  My Brother told me he wanted to take the fully restored Keaton to the Memorial.  It was Winter, river running high, water very cold.   We hit the river and right when we got to the memorial the engine started overheating.  In front of a few hundred people at the memorial, nearly all were Keaton Boat fans, we had to lift the engine cover and start figuring out what was wrong.   Keep in mind he just rebuilt everything.

So I find the leak in a very difficult to get to place under the engine, and the hose I found leaking was obviously years old and rotten.  No way to do a temp patch.  This hose is the Main water supply to the engine from the jet pump.  Me being ever compassionate looked at him and said, You Told Me You changed all the hoses?   He looks right back at me and says, I did, but that one is hard to get to.   Like that explained everything.

This beautiful Keaton had to be towed back by two very nice fishermen in a 15 foot aluminum boat with a 10 HP outboard.  Of course the launch was upstream so it took forever.  Very Happy they were there to assist.  

The river would have had us a mile downstream before we could have paddled to shore.   

Many Keaton enthusiast's at Jack's memorial got a laugh at us and a few actually offered help.  I wanted to crawl under the deck and hide.  Due to the boat issue we missed the memorial completely but the engine was shut down in time and had no damage other than the broken hose and the exhaust hoses collapsed which helped us recognize the water supply issue quickly.