In the film "Hidalgo", set around 1890, there's a scene where the
protagonist gets into a fistfight on a transatlantic voyage. A friend
in first class gives him some ice to apply to his face, but the friend
says that ice is a rare commodity at sea, and complains about having
to drink his gin warm.

My question is, did they have the technology to "make" ice in 1890 on
an ocean liner? Or did they just load ice on board when at port, and
if so how did they prevent it from melting? Would they have had ice
cubes, or just one big block that they had to chip away at?

Actually, making ice cream is basically a very easy process that does
not need any sophisticated equipment at all. In the mid-19th century,
maybe even earlier, ice cream was made with small portable "machines"
that used the low temperature that results from the reaction of salt
in water. Constant stirring (with a mechanism manually powered via a
crank) kept the flavored milk creamy while freezing.

Scriptor

Although mechanical refrigeration existed in 1890, I doubt that it
would have been in use on a ship.

Apparently riverboats in the early 20th century were still using
blocks of ice for keeping food cool:

"An ice house, south of the commissary, stored large blocks of ice
buried in sawdust. The ice was used on the riverboats to refrigerate
perishable foods."

http://www.yukonweb.com/business/lost_moose/books/edge/shipyards.html

Apparently mechanical refrigeration was used onboard ships in the 19th
century when the cargo required it:

"James Harrison, a Scottish journalist, moved to Australia from
Glasgow and set about designing his own refrigeration machine.  In
1855 he succeeded in creating and patenting an ether liquid-vapour
compression fridge.  The compressor worked by exerting pressure on a
refrigerant gas, forcing it to pass through a condenser, where it
cooled down and liquefied. The liquid then circulated through the
refrigeration coils and vaporized again, cooling down the surrounding
air.

Australia was in desperate need of refrigeration because of its lack
of natural ice needed for keeping food cool.  Harrison was convinced
that the economic salvation of Australia lay in the marketing of her
millions of sheep and cows to the millions of Europeans.  But without
refrigeration, it was impossible to ship the mutton across the 100
Degree-plus Equator.

Harrison spent his last penny to equip the Norfolk ship with a
chemical freezing mixture for twenty tons of beef and mutton at
Melbourne.  But when the meat arrived in London, it was discovered
that the chemical tanks had leaked and ruined the entire cargo.  As a
result, James Harrison went into bankruptcy, even being forced to sell
his successful newspaper business.  Though Harrison was financially
devastated, he did open the door for the economic salvation of
Australia.  Other successful refrigerated voyages followed, which
finally convinced Europe that Australia had something to offer, and
that frozen food could be both safe and delectable."

http://www3.bc.sympatico.ca/st_simons/cr0003.htm

scriptor's astute comment took me back.  Any kid who once attended a
church ice cream social in the South on Sunday afternoons knows about
turning a hand crank for extended periods.  Another kid would sit on
top of a blanket on top of the churn and add salt.  Fantastic ice
cream and fun!  Flavors were limited only by vanilla extract and the
various fruits one had picked in the garden.  No extension cords
required---just muscles.  Those were the days!

methusaleh ron

Scriptor's comment just confused me, since the question said nothing
about ice cream, but only ice itself.  But since we're on the subject,
Thomas Jefferson was said to have greatly enjoyed ice cream, though I
don't have the details on how he made it.