Aragorn was captured in Moria and tortured (maybe)
One possible explanation for Aragorn's attitude towards the prospect of re-entering Moria, maybe the only explanation with any support in any of the texts, is that Aragorn had been captured in Moria and tortured.
In early drafts of The Lord of the Rings, the Ranger who helps Frodo reach Rivendell and then joins the Fellowship of the Ring was nicknamed not "Strider" but "Trotter". Indeed, he had the nickname "Trotter" before Tolkien gave him the name "Aragorn". At one time, this character was a "wild" Hobbit who wore wooden shoes.
Aragorn started as "Trotter", a Hobbit with wooden shoes:
Suddenly Bingo noticed that a queer-looking, brown-faced hobbit, sitting in the shadows behind the others, was also listening intently. He had an enormous mug (more like a jug) in front of him, and was smoking a broken-stemmed pipe right under his rather long nose.
He was dressed in dark rough brown cloth, and had a hood on, in spite of the warmth, - and, very remarkably, he had wooden shoes! [...]
'O! that is one of the wild folk - rangers we call 'em. He has been coming in now and again (in autumn and winter mostly) the last few years; but he seldom talks. [...]
What his right name is I never heard, but he's known round here as Trotter. You can hear him coming along the road in those shoes: clitter-clap - when he walks on a path, which isn't often. Why does he wear 'em? Well, that I can't say. But there ain't no accounting for East or West, as we say here, meaning the Rangers and the Shire-folk, begging your pardon.'
— isanae, Science Fiction & Fantasy Stack Exchange answer #150458, apparently quoting a draft of The Lord of the Rings published in The History of Middle-earth
The emergence of Aragorn as a Man is covered in chapter 1 of History of Middle-earth 7, Gandalf's Delay, and I'll quote the note in full:
Trotter is a man of Elrond's race descendant of [...] the ancient men of the North, and one of Elrond's household. [...] [> Gandalf and Bilbo] arranged with Trotter (real name [...] Aragorn son of Aramir) to go towards the Shire and keep a lookout on the road from East (Gandalf was going South). He gives Aragorn a letter to Frodo. Aragorn pretends he is a Ranger and hangs about Bree. (He also warns Tom Bombadil.)
Reason of wooden shoes - no need in this case because Aragorn is a man.
In other plot notes he was an Elf, and Tolkien even once reverted to him being a Hobbit before finally and definitively settling on him as a Man. [...] The name "Trotter" survived until quite late in the writing of Lord of the Rings, interestingly enough.
— "user8719",¹ Science Fiction & Fantasy Stack Exchange answer #86229, quoting notes of Tolkien's published in The History of Middle-earth
From the first known draft of this scene, Trotter shows a fear of Moria and, very similarly to the published version of Aragorn, says that "the memory is evil":
In the very first draft of the chapter, the hobbit Trotter (who would eventually morph into Aragorn), says:
'Yes, I know of the mines,' said Trotter quietly. 'I went there once, and the memory is evil
History of Middle-earth VI The Return of the Shadow Chapter 24: "The Ring Goes South"
A margin note indicates that Trotter was captured there by the forces of Sauron, but no details are given.
— Jason Baker, Science Fiction & Fantasy Stack Exchange answer #106831
Trotter the Hobbit wore shoes because of chronic pain that he suffered in his feet after being tortured by the enemy after he ventured too near to Mordor ("past the Dead Marshes") while tracking Gollum:
Thus it was that Frodo learned how Trotter had tracked Gollum as he wandered southwards, through Fangorn Forest, and past the Dead Marshes, until he had himself been caught and imprisoned by the Dark Lord. 'Ever since I have worn shoes,' said Trotter with a shudder, and though he said no more Frodo knew that he had been tortured and his feet hurt in some way. But he had been rescued by Gandalf and saved from death.
The Return of the Shadow - "In the House of Elrond"
Eventually Tolkien decided to make Trotter a man and cut out the whole arc with the wooden shoes. And then sometime after that Trotter was renamed to Strider.
— "ibid",² Science Fiction & Fantasy Stack Exchange answer #254450
Even as a Man³ rather than a Hobbit, when there no longer was a need to justify his wearing shoes, Trotter still had been tortured, and his torment was noted to have followed his being "caught in Moria", with no mention of tracking Gollum:
Again from Gandalf's Delay:
Trotter is a Ranger - descendant of Elendil? - he is known to Bilbo, and Gandalf. He has previously been to Mordor and been tormented (caught in Moria). Gandalf brought him back towards borders of Shire in April. [...]
— "user8719", Science Fiction & Fantasy Stack Exchange answer #86229
I don't have access to the texts to be sure, but it seems plausible that it was because the Trotter quoted by Jason Baker associated entering Moria with being captured and tortured that he reacted fearfully to the possibility of re-entering Moria and described his prior experience in Moria in as obscure a way as "the memory is evil", and it seems plausible that Aragorn's "very evil" memory would be similar to that Trotter's "evil" memory.
I admit that the reference to April in the "caught in Moria" passage does not seem to fit the timeline of the published Lord of the Rings and thus makes clear that these events translate, if at all, only in some adapted form into the published story. Perhaps no explicit confirmation can be found that the published version of Aragorn, as opposed to Trotter, ever was captured and tortured — that Jason Baker, who has much more knowledge of, and access to, Tolkien's work than I, found so few details for his answer to this question (answer #106831) does suggest that such confirmation cannot be found.
Still, we also have (so far as I am aware) no confirmation that Aragorn did not suffer the same torture as Trotter, and I think Aragorn's aversion to the prospect of re-entering Moria and his seeming preference to refer to his experience only vaguely ("the memory is very evil") seem consistent with associating Moria with a highly unpleasant ("very evil") memory of being tortured.
¹ "a deleted account for a Tolkien expert", according to Ian Thompson
² That's a user named "ibid", not the citation abbreviation "ibid.".
³ Elendil is a Man.