TL;DR: Symbiotes' names when they're just black Klyntar goo are unclear. As for when they're bonded, it's unclear whether their names are "inherent" or resulting from the symbiosis - save for Venom, which was a deliberate choice.
While on Klyntar/not bonded
In the early appearances of the Venom Symbiote, it wasn't referred to with a name ("the alien costume", "the Symbiote", etc). This might be a matter of the writers not having a definite idea, or just a storytelling perspective, but later on when the Symbiote lore grew more consistent, the Symbiotes were still not referred to with names of their own.
For instance, when Knull, the "God-Host", addresses Eddie's Symbiote in Venom #4 (2018), he always calls him "my child". While it could be argued that it's only him being condescending, he does address Eddie both as "host" and by his civilian name.
Furthermore, Symbiotes operate in a hive mind. Without dwelling into the various "hive mind" definitions, most of the time it strips its members of individuality. If all your thoughts are connected/shared, does one need a name?
Although it might be possible they do have names, just not "as other civilizations would understand it", as it's the case with their culture (Venom #3, 2017). Such names could be understood by hosts, though (see Toxin example below).
In Venom's big reveal in The Amazing Spider-Man #300 (1988), Eddie gave a little backstory of what he became after Spider-Man proved a big article of his wrong, which led to to Eddie being fired. As one still needs to pay the rent, Eddie wrote for cheap newspapers, with lurid "I was abducted by aliens" articles and the like. For a journalist at heart, that counts as spewing out venom, hence the name.
In-universe, Pat Mulligan (human part of Toxin Symbiote) refers to Carnage as a Symbiote entering Cletus Kasady. This might imply that Carnage's name was already is own, but as he says the same of Venom, it's possible he's just mistaken and thinks that the Symbiotes' names are their own - Venom's wasn't, as explained above. Panel is from Toxin #1 (2005).
Out-of-universe, Carnage's name was a bit of a debate, as revealed in an interview given by Danny Fingeroth (former Marvel Comics editor) to Syfy. Some names which would fit a psycho killer like Carnage were already taken, such as Chaso and Ravage. The writers settled on Carnage.
Prior to settling on the name Carnage, Fingeroth said, the creators at Marvel cycled through several other names, including Chaos and Ravage, which were in use elsewhere in the Marvel universe.
Marvel creators reveal Carnage's sociopathic origin, plus what's next for Venom, Syfy, October 19, 2018
This one is self-explanatory. Eddie "purged" the Venom Symbiote out of his body, but some remnants still stayed, and when he underwent a shady cancer cure, those were turned into all-healing powers cells. Especially, they meant to cure the "poison, the disease" Venom was, as shown in The Amazing Spider-Man #569 (2008). What a better name than Anti-Venom?
Toxin was given his lovely name by his grandfather, Venom, in Venom vs. Carnage #2 (2004). How Venom came up with Toxin instead, of, say, Little Bobby Tables, seem pretty obvious.