O.k., here's a long one, with my conclusions on the tune's origins at the bottom. (FWIW! and Whew!)

To provide more detail on the possibly American Revolution origins of the tune (and to expand on Guy Wolff's original 1998 post (!!) about a possible connection to Marblehead, Massachusetts), here is information from Wikipedia on the Battle of Long Island, also known as the Battle of Brooklyn:

"After defeating the British in the Siege of Boston on March 17, 1776, General George Washington, Commander-in-Chief, brought the Continental Army to defend New York City, then limited to the southern end of Manhattan Island. There he established defenses and waited for the British to attack. In July the British, under the command of General William Howe, landed a few miles across the harbor on Staten Island, where they were slowly reinforced by ships in Lower New York Bay over the next month and a half, bringing their total force to 32,000 men."

"On August 22, the British landed on the western end of Long Island, across The Narrows from Staten Island, more than a dozen miles south from the East River crossings to Manhattan. After five days of waiting, the British attacked American defenses on the Guana (Gowanus) Heights. Unknown to the Americans, however, Howe had brought his main army around their rear and attacked their flank soon after. The Americans panicked, although a stand by 400 Maryland troops prevented most of the army from being captured. The remainder of the army fled to the main defenses on Brooklyn Heights. The British dug in for a siege but, on the night of August 29–30, Washington evacuated the entire army to Manhattan without the [further] loss of material or a single life. Washington and the Continental Army were driven out of New York entirely after several more defeats and forced to retreat through New Jersey and into Pennsylvania."

Details:

"Washington and the army were surrounded on Brooklyn Heights with the East River to their backs. As the day went on, the British began to dig trenches, slowly coming closer and closer to the American defenses. By doing this, the British would not have to cross over open ground to assault the American defenses as they did in Boston the year before. Despite this perilous situation, Washington ordered 1,200 more men from Manhattan to Brooklyn on August 28. The men that came over were two Pennsylvania regiments and Colonel John Glover's regiment from Marblehead, Massachusetts."
...
"Washington sent a letter instructing General William Heath, who was at Kings Bridge between Manhattan and what is now the Bronx, to send every flat bottomed boat or sloop without delay in case battalions of infantry from New Jersey might come to reinforce their position. At 4:00 p.m., on August 29, Washington held a meeting with his generals. Mifflin advised Washington to retreat to Manhattan while Mifflin and his Pennsylvania Regiments made up the rear guard, holding the line until the rest of the army had withdrawn. The Generals agreed unanimously with Mifflin that retreat was the best option and Washington had orders go out by the evening.

"The troops were told that they were to gather up all their ammunition and baggage and prepare for a night attack. By 9:00 p.m., the sick and wounded began to move to the Brooklyn Ferry in preparation for being evacuated. At 11:00 p.m. Glover and his Massachusetts troops, who were sailors and fishermen, began to evacuate the troops."
...
"At 7:00 am, the last American troops landed in Manhattan. All 9,000 troops had been evacuated without a single life lost."
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"When news of the battle reached London, it caused many festivities to take place. Bells were rung across the city, candles were lit in windows and King George III gave General Howe the Order of the Bath."

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Other Revolutionary era activity on Staten Island:

"… on September 11, 1776, the British received a delegation of Americans consisting of Benjamin Franklin, Edward Rutledge, and John Adams at the Conference House on the southwestern tip of the island (known today as Tottenville) on the former estate of Christopher Billop. The Americans refused the peace offer from the British in exchange for the withdrawal of the Declaration of Independence, however, and the conference ended without an agreement."

There was also a Battle of Staten Island in 1777, but this was an unsuccessful raid by the Americans on the British forces.

British forces remained on Staten Island throughout the war, and they used the island as a staging ground for their final evacuation of New York City on December 5, 1783.

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A few other notes on my research on the tune:

The tune does not show up in Bruce Olsen's "Incomplete Index of Scottish Popular Song and Dance Tunes Printed in the 18th Century".

Perhaps Jack can check Charles Gore's 'The Scottish Fiddle Music Index'.

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I'll take my turn at informed speculation:

That James Robertson (music notebook, Scotland, 1796) includes it along with other tunes that come from the American Revolutionary period suggests strongly that it's Staten Island, New York, not Staten Island, Cape Horn, Argentina.

If the tune has any relation to these events, it was most likely written by someone on the British side of the conflict. I don't think hornpipes get written for defeats.

However, as defeats go, a sneaky and successful retreat is worth some celebration, and as Guy, the OP, suggests, the Marblehead, Massachusetts, troops played an important role, that might be worth celebrating. On the downside for this theory, why call it "Staten Island Hornpipe" then, if that where the British were stationed, and the successful action was elsewhere? Irony? That's a bit of a stretch.

That the tune found itself in print in Britain only six years (Aird's Airs, book 2, Glasgow, Scotland, 1782) after the Battle of Long Island suggests strongly that the tune was written either in association with the celebration in Britain after the battle, or by someone who was among or associated with the British troops stationed on Staten Island around that time.

Any connection to the Irish on Staten Island, NY, would be after the fact.

A useful data point that would raise the odds for the Cape Horn Staten Island would be if there any publications of the tune prior to 1776. However, I have to ask, how much ship traffic would there have been on that route prior to 1776? European and American whalers didn't go there until 1788.

The tune could be found in the United States at most 100 years later and has been in active use here for more than 100 years.

I'm curious if there are any American printings before Ryan's Mammoth Collection (like earlier, from Elias Howe, or earlier than that).

Here's where I'm putting my money today: My bet is on the connection with the American Revolution, and that the tune was written by someone on the British side, probably in Britain, and probably someone with a connection to Scotland (like a Scot!).

~ Becky in Tucson