There's no telling with certainty whither Carroll may have wandered, but the contemporary sense of 'boiling an ocean', that is, "to undertake an impossible task or project or to make a task or project unnecessarily difficult" has no apparent connection to the much-anticipated Walrus and Carpenter's talk about "why the sea is boiling hot".
Oceans, or at least parts of them, after all, are routinely made boiling hot. Making oceans boil is neither "impossible" nor especially "difficult". Volcanoes and underwater geothermal vents naturally boil the ocean; nuclear reactors and industrial flumes boil the ocean artificially.
A search for the origin of the contemporary sense of "boiling an ocean" did not produce anything definitive; however, the earliest mention of that sense of the phrase that I found, from 1986, attributed it to Will Rogers:
Balanced budget: When the seas boil
....
It all reminded him, he said, of what comedian Will Rogers replied when he was asked during World War I what he would do about the threat of German submarines in the North Atlantic.
His response was that he would set the oceans of the world to boiling and thereby cook anybody who happened to be in a submarine in the coean," said DeConcini.
"But how do you get the oceans to boil?" Rogers was asked.
The comedian replied that he had absolutely no idea, but that it would certainly solve the problem if they could be made to do so, the senator recounted.
"Well," said DeConcini, "we did not boil the oceans, the sun does not rise in the West, and members of Congress will not bring about balanced budgets if all we have to rely on is their best efforts. ... We cannot boil oceans and we cannot balance budgets unless we have a different legal framework in which to legislate."
The Marshall News Messenger, Marshall, Texas, 1 Apr 1986 (paywalled; emphasis mine).
The question of why the sea is boiling hot, however, may have an answer or two. Those answers, in turn, may have provoked Carroll to bring the question up. The most likely account, in my estimation, comes from northern European mythologies. While Carroll almost certainly would have been familiar with these mythologies, in one or more forms, a Finnish version summarized in 1898 in an article in The Westminster Review titled "The History of the Forms and Migrations of the Signs of the Cross and the Su-astika" proved most readily accessible to me:
...the cult of the alligator. This was originally in Finn mythology the sun-lizard, the child of the mother goddess Kesari-tar, the daughter (tar) of the kettle (kesari), who was the goddess of the burning mountain containing the heat generated in the lower earth by the revolutions of the Northern Pole, the invisible fire-drill of heaven turned by the stars of the Great Bear, and not by the ape-god of Southern mythology. She was made pregnant1 by the froth of the sea heated to boiling-point; and the father god, who made the fire-drill to revolve and boil the ocean water, was Il-marinen, the Northern Smith, the Finn form of the German Wieland the smith.
- Abercromby, "Magic Songs of the Finns", Folk-Lore, vol. i pp. 381-2.
Although that particular account was published in 1898, twenty-seven years after the publication of "The Walrus and the Carpenter", such striking details of the substance of the northern European mythologies had probably not escaped Carroll's notice.
Another possibility, and one that especially might have provoked a desire for the discussion of a Walrus and a Carpenter (although the latter was so-named by the choice of Tenniel from the other options of a butterfly and a baronet), springs from Christ's description of Judgement Day. As described in a sermon by the Rev. Dr. Cahill, published in 1882 in A Treasury of Irish Eloquence,
It is He Himself that will, on the terrible day, boil the oceans with His angry breath....
This source, again, post-dates the publication of "The Walrus and the Carpenter", and would at most suggest a general rather than a specific origin of the Walrus's proposed topic for discussion.
Sources antedating the publication of the poem also surface in a thorough search but, as with the later sources, only suggest what might have been rather than establishing what was the origin of the Walrus's proposal. For example, a quasi-mythological passage from correspondence addressing "Liquid Fuel" in the 24 Apr 1868 Journal of the Society of Arts might have been a specific source:
We, with our limited means, cannot "set the Thames on fire;" but Nature, with her unlimited means and her huge central furnace, may convert water, on a large scale, into its constituent gases...nothing disappears in one form but it comes forth again in another. We have lost the dodo, but we have the duck still, and the elephant replaces the megatherium. The large animals grow fewer in number, but the smaller races multiply. That serpent that stopped the army of Regulus in Africa is no more, and the sea serpent is problematic; but the Cachalot whale still bursts in ships' sides in his ire, with his huge head, and volcanoes break forth and disappear, and, if later speculators do not err, boil the ocean into the Gulf Stream, and render northern countries habitable by the human race....
The gist of these considerations is that, unless and until Carroll's missing diaries and diary pages turn up and explain the mystery, we cannot know whence sprang the topic suggested by the Walrus. By the same token, the inspiration for Will Roger's putative suggestion for surfacing subs may remain unknown, and along with it any direct or indirect connection the latter may share with Carroll's bon mot.
