So, taking a look at the Basic D&D rulebooks...
Basic 1977
No monsters appeared as core here and nowhere else. (Basic '77 was kind of an odd duck anyway, more of an intermediate step between 0e and 1e.)
Basic 1981
Just one that didn't go onward at all - the medium, which was just a low-level NPC wizard. One presumes it was folded into the more general wizard NPCs in later editions, to include the 5e mage. (Though if you want an actual novice wizard in 5e, you have to go for the apprentice wizard in Volo's Guide to Monsters.) That said, its Basic D&D counterparts - the veteran (fighter) and acolyte (cleric) - made it into the current edition's core, so make of that what you will.
There were also many introduced here that stuck around through Basic '83 and the Rules Cyclopedia, but never moved onto the AD&D core:
- Living statues (in crystal, iron, and rock flavors)
- The legendary thoul
- White ape
- Rock baboon
- Robber fly
- Cave locust, later known as giant locust
- Basic's version of the giant bat (which was a bit different from the contemporary AD&D version, in that they had a vampire angle)
- Giant ferret and giant shrew
- Two types of giant beetle: oil and tiger
- Four types of giant lizard: draco, gecko, horned chameleon, and tuatara
- Three types of giant spider: black widow, crab, and tarantella
- Four types of snake: pit viper, giant rattler, rock python, and (normal-sized) sea snake
Some of the above did make non-core comebacks in the AD&D line, most notably in the 2e Mystara MC Appendix. Living statues have been the most popular to revive, having appeared in that plus 3e and 5e sources.
I'm not sure what these unique core choices say about the Basic D&D ethos. Maybe they were trying harder to make the low-level dungeon crawling experience interesting? They certainly felt empowered by 1981 to forge their own path of what should be in the core, in any case, such as going with "living statues" rather than golems. (Although Basic did later introduce its own distinctive golems.)
As for why any of these didn't carry forward, not sure. Much weirder things have gone the distance. Even the 3e Monster Manuals, which were clearly trying to draw on Basic D&D for ideas, skipped these older creatures in favor of Rules Cyclopedia options.
Basic 1983
Not much here that wasn't in Basic '81, but one exception was the giant racer snake. It did carry forward into the Rules Cyclopedia, but made no other appearances. Here's its '83 description:
Why did it get added to the other snakes? No idea. Maybe someone thought racer snakes were cool?
My guess as to why it didn't make it into other editions? It probably wasn't distinctive enough compared to other snakes.