There's a popular brand of rum called Captain Morgan's Spiced Rum.  I
seem to remember reading somewhere that Captain Morgan was a real sea
captain, not exactly a "pirate" but at least some kind of adventurous
seafaring type.  What's the truth?  Who was the guy?  What was his
ship named?  Did he do anything notable?  And how on earth did he get
rum named after him?

Captain Morgan (1635-1688) was a buccaneer Welshman who won an English
knighthood for his attacks on Spanish colonies and Panama.  His attack
on Panama took place just after peace was declared between Spain and
England, so he was arrested.  He was later released (when things got
rocky between Spain and England again) and became the Deputy Governor
of Jamaica.

Interesting history is here:
http://www.interknowledge.com/bahamas/morgan.htm

Also here:
http://www.mesa.k12.co.us/pirates/morgan.html

Morgan had a whole fleet of ships at his command, and I couldn't find
record of all their names, but his flagship was the Oxford.  It
exploded under curious and unexplained circumstances, killing most of
his crew.  Needing another ship, he stole one (!) from called the Cour
Volant a French pirate, and renamed her the Satisfaction.

As for the rum issue, remember that Sir Morgan lived in Jamaica. 
After retiring as Deputy Governor he lived on his Jamaican estate and
grew sugar cane  which he used to make rum. He was reputed to be quite
a drinker, with a report from Lord Vaughan, Governor of Jamaica, from
1674 that claimed he "frequented the taverns of Port Royal, drink-ing
and gambling in unseemly fashion." (This comes from the site listed
above).  Whether the Captain Morgan brand can be traced back all the
way to Sir Morgan's original distillery, I'm not sure.  The brand may
just pay homage to this fascinating fellow.  Currently the brand is
owned by Seagram, which was until fairly recently based out of Canada,
and is now based in the US.

If you want to research this a bit more, or just check out Captain
Morgan rum, you can visit the Captain Morgan site at
http://www.rum.com .

There is also an interesting article about a dispute over the Captain
Morgan brand in this article at the BBC News site:
http://news.bbc.co.uk/1/hi/business/1068392.stm

I hope this was useful to you.  Truly, Captain Morgan seems to be
quite a remarkable man.

That is an excellent answer, exceeding my expectations.  Thank you for
indulging my curiosity so thoroughly!

There are a number of movies about Captain Morgan ... 

- - - Pirates of Tortuga - 1961, Ken Scott, John Richardson: Sir Henry
Morgan's exploits.

- - - Morgan the Pirate - 1961 (Italian), Steve Reeves, Valarie
Lagrange: Great tavern scenes, and real ships.

 from Yer Piratical Movie Guide at
http://planetrory.8k.com/misc/piratical.htm

 . . . but the one I've seen is: The Black Swan (1942) with Tyrone
Power as Morgan's right-hand man and Maureen O'Hara as a reluctant
love object. Wonderful old romance adventure.

   "This sweeping, beautifully made pirate epic has Power play a
swashbuckling aide to the notorious buccaneer Henry Morgan, who is
pardoned from the gallows and is sent to Jamaica as the new governor.
Tyrone falls in love with the gorgeous Maureen O'Hara, the daughter of
the previous governor - who spurns the advances of this rough-hewn
adventurer...The dialogue and story is very reminiscent of the earlier
Flynn vehicle CAPTAIN BLOOD, and although the film is a trifle stilted
in capturing the idiom of the period, the action is overwhelming once
it begins...."
http://simplest-shop.com/The,Black,Swan--1-162026-6303943004-0-video-product.html

More info: 
Some pirate films sail on forever 
http://www.augustachronicle.com/stories/050203/uhl_220-7397.000.shtml