For one reason or another, this question had gnawed at me over the night, so I decided to look into it. I'll give you the full account just for posterity and your entertainment.

I originally found a few stories referring to a blind horse named "dick" that fell down the road that runs next to the mountain, thus he was the "dick" that "tumbled-down". However, this same story appears in reference to mountains with the name in both Maine and New Hampshire. Given the similarities in wording and a commonly mentioned reference to a forgotten connection to a Native American phrase, this 1889 history of Carroll County, New Hampshire appears to be the source of this narrative. There certainly could have been a horse named "Dick" that took a tumble, but it seems a bit far fetched given the Cromwell connection.

I then did a search on old New England newspapers to see if I could find any hits there. The only real link was an article in The Providence Gazette on April 28, 1781. On page 3, a small recount of events earlier in the year on Sint Eustatius, an island just north of St. Kitts and controlled by the Dutch. The article mentions Tumbledown Dick Bay was the site for fighting between the British and Dutch due to the Dutch using the island to supply the Americans with arms throughout the American Revolutionary War. I won't go into much more detail as there is a solidly written wikipedia article on the affair. I had wondered if the naming of the bay after the ill-fated Cromwell son was a method of the Dutch thumbing their noses at the British, and that the colonists in Northern New England were simply doing the same. This seemed like a fairly unlikely theory though. Attempts to find examples of "tumbledown dick" as an American method of mocking the British did not yield much of anything. This ended up being a Caribbean goose chase.

A return to the original search then yielded the answer that seems most likely. It appears in a book that is perfectly suited to answer the original query.

Christopher J. Lenney, Sightseeking: Clues to the Landscape History of New England, (Durham, NH: University of New Hampshire Press, 2003), 25.

I'll quote the entire paragraph as I fear context will be lost by pulling apart individual sentences. Link on book title will send you directly to the page in question. Emphasis mine.

There is a curious similarity between some of the more blatantly picturesque hydrographic toponyms and the traditional Anglo-American names for taverns and coffeehouses. The inspiration for such names might have come directly from the signboards of such places of seafarers' resort, or coincidentally from some common source of medieval fold imagery that also fond expression in coats of arms. In all these contexts, it was the iconic and mnemonic power of the image that recommended its usage. The Rose and Crown is a boot-shaped shoal off Nantucket charted as early as 1675. While the name of no know British danger, the Rose and Crown since at least 1606 has been a perennial and prolific Lincoln signboard; it derives from a badge of the Tudors. (It was also not unknown as a ship name.) Hen and Chickens, a marine toponym common to old and New England, is a popular London tavern name (known also from Burlington MA), and incidentally is also tavernkeepers' slang for pewter pots of mixed sizes. Labo(u)or-in-vain is a creek in Ipswitch MA; the Sugar Loaf designates each of twin islands in Vinalhaven ME, as well as many mountains; Tumbledown Dick (a supposed reference to the short rule of Richard Cromwell) is a headland, a rock, and a mountain in Maine; Hedgehog is the name of a Maine island and numerous hills and mountains. All are London signboards.

After all this work, it seems as though these landmarks are just named after the bar, which has existed since the 1600, but since been turned into a McDonald's.