I hate this meme. It captures how so much of Tolkien’s subtlety and depth can go whooshing over the heads of many readers (or, more likely, viewers). ‘Mount Doom’ was not chosen to be vaguely menacing, like some comic book villain calling himself ‘Dr. Doom!” Tolkien uses the word ‘doom’ in a very particular (and very old) sense, which has huge symbolic significance for the story as a whole.
Put as simply as possible, ‘doom’ means both ‘fate’ and ‘decision,’ sometimes even simultaneously, not necessarily with a negative connotation. The verb form still retains the meaning of ‘decide’ as deem, though the noun has mostly come to mean other things. Only Master Elrond can pull off using both the noun and the verb in the same sentence, as in, “That is the doom we must deem.” Do not try that at home.
Tolkien riffs on the dual meaning of ‘doom’ as a way to represent the interplay of fate and free will. This is seen most clearly in the story of the self-styled ‘Master of Doom’ himself, Túrin Turambar (Don’t panic! This is the only paragraph about The Silmarillion!). Túrin is constantly aware that a not-so-pleasant doom is close on his heels. He tries everything to escape his doom, he insists that he is the master of his own fate, and yet every decision he makes drives him closer to it.
Frodo doesn't try to run from his doom, but walks into it with open eyes. At the Council of Elrond, Frodo very clearly chooses to take on the task; his agency in that decision is emphasized in both the book and the movie. No one just assumes that he will take the Ring south. Yet almost immediately, Elrond seems to insist that it was fate all along: “If I understand aright all that I have heard, I think that this task was appointed to you, Frodo, and if you do not find a way, no one will.” (The same words are delivered by Galadriel in the movie.)
Consider also how Sam begins his brief stint as Ringbearer. Another author might have taken it for granted that Sam would take up the ring. But Tolkien dwells on the moment, making it abundantly clear that it was a choice. Before deciding to continue the journey, Sam considers pursuing Gollum for vengeance or committing suicide as alternate options.
The Mountain is clearly a symbol of the doom that Frodo and Sam must face. Both of them willingly choose to go there; it is a destiny that they cannot escape. So what better name could there be for it than Mount Doom?
Fun fact: In both of the chapters I mentioned – “The Council of Elrond” and “The Choices of Master Samwise” – there is an explicit reference to Túrin Turambar. Coincidence?