Which “classic” Dungeons & Dragons monsters were truly the original ideas of the game’s creators, as opposed to being adaptations from existing literature or folklore?

The original 1977 Monster Manual for 1st edition Dungeons & Dragons is a mix of ordinary animals (Bears, Camels, Jaguars, Mammoths, etc.), mythical critters from legend (Centaurs, Lamia, Naga, Pegasus, etc.) - including non-European legends, as critters like the Efreet and Roc were drawn from Middle Eastern folklore and the 1,001 Nights - as well as entities borrowed from or inspired by contemporary fantasy fiction - the D&D troll, for example, is inspired fairly directly from the trolls in Poul Anderson's Three Hearts and Three Lions (1961), the vampires are inspired by Bram Stoker's Dracula and its subsequent media adaptations, the Elves, Orcs, and Dwarfs borrow more from Tolkien's The Lord of the Rings (1955) trilogy than folklore. Because of copyright restrictions with the latter, there are some sly variations as well - the D&D Treant is basically a Tolkienian Ent, the D&D Halfling is basically a Hobbit - D&D ended up avoiding using specific terms after a threatened lawsuit by the Tolkien Estate.

You could argue that some of the most distinctive monsters are really variations on a theme - Dragons for example, were a staple of myth, and the specific fire-breathing, flying lizard form was immensely popularized by The Hobbit; what D&D adds is variety (different colors, different breath weapons/abilities, etc.) Some of this could be taken as an expansion of the material that Tolkien had already produced in The Hobbit and the Silmarillion (1977), since Tolkien writes of "cold drakes" and flightless dragons, but many of the specific developments characteristic of these first D&D dragons are really original - metallic for good dragons, colors for evil dragons, etc. - even if they took inspiration from previous fiction.

However, some of the monsters are truly unique and unprecedented, or at least unique takes on classic monsters. When D&D in it's third edition went with an Open Gaming License, the following critters were deemed part of D&D's "product identity":

Beholder
Gauth
Carrion crawler
Displacer beast
Githyanki
Githzerai
Kuo-toa
Mind flayer
Slaad
Umber hulk
Yuan-ti

Many of these critters weren't in that original Monster Manual, but were developed later on and became part of the game line. Fantasy and science fiction author Charles Stross actually created the Githyanki, Githzerai, and Slaad as a teenager, where they appeared in the Fiend Folio (1981).

Some of the critters, while original and unique to D&D, nonetheless have inspirations from pulp and paperback fiction of the 1930s-1970s. Gary Gygax for example wrote in an interview:

The mind flayer I made up out of whole cloth using my imagination, but inspired by the cover of Brian Lumley's novel in paperback edition, The Burrowers Beneath [1974].

Likewise, the Kuo-Toa have gross similarities to H. P. Lovecraft's Deep Ones, the Yuan-Ti to Robert E. Howard's Serpent People, etc.

The bulette and rust monster also owe a debt to a bag of cheap plastic Prehistoric Toys which Gygax used as miniatures in his first games (keeping in mind that D&D started out as a miniatures war game and evolved into a roleplaying game). Playtester Tim Kask recalled:

There once was an unknown company in Hong Kong that made a bag of weird animal-things that were then sold in what once were called dime stores or variety stores for like $.99. I know of four other very early monsters based on them. Gary and I talked about how hard it was to find monster figures, and how one day he came upon this bag of weird beasts…He nearly ran home, eager as a kid to get home and open his baseball cards. Then he proceeded to invent the carrion crawler, umber hulk, rust monster and purple worm, all based on those silly plastic figures. The one that I chose was known in the Greyhawk campaign as “the bullet” (for it’s shape) but had only amorphous stats and abilities, not being developed. Gary told me to take it home, study it, and decide what it was and what it could do. The bullette (boo-lay), as it was first called, was the first monster I invented. Why is the more interesting part of the story. I had decided to add a feature to DRAGON that would mean a new monster every issue; problem was, I had to launch an issue early because an ad didn’t come in. I wrote it up very late at night; the nickname “landshark” was a reference to a character that the original Not Ready for Primetime Players had done on Saturday Night Live. I went to Dave Sutherland for an emergency drawing (drawings could be submitted to the printers after the copy was set) and he did a dandy job on almost no notice.

Other critters are more obscure in origin. The Ixitxachitl for example, a kind of evil, intelligent manta ray, first appeared in David Arneson's Blackmoor (1975); and if it has a prior fictional inspiration I'm not aware of it. The same goes for the Beholder, which first appeared in Greyhawk (1975) and was created by Theron O. Kuntz.