Okay, I know it's common knowledge here that Teucer shares his name with King Teucer, who ruled over the region that would later become known as Trojan territory. That he's named after a figure from Trojan history isn't strange, since his mother, Hesione, was a Trojan princess and Priam's sister. The interesting part is that the Homeric scholia says this:
...you, though a bastard child (καί σε νόθον περ᾿ ἐόντα)
When Heracles sacked Troy, he took as prisoner Hesione, the daughter of Laomedon (and sister of Priam), and gave her as a war-prize to Telamon because he had fought with him. Telamon fathered Teucros by her. Because he had been born of a Trojan woman, they called the child “Teúcros,” moving the accent back to create a proper noun. For the Trojans are called Teucroí after one of their rulers, Teucros. The story has been told in more detail by many, including Apollonios the grammatikos in the second book of On Generations.
Translation by R. Scott Smith et al.
In other words, in this version told here by the scholia, Teucroi/Teucrians would be a kind of synonym for Trojans because of King Teucer/Teucros. The scholia even uses the term as a synonym here:
[a bane] unto himself, since he did not know the gods’ decrees ([κακὸν...] οἷ τ᾿ αὐτῷ, ἐπεὶ οὔ τι θεῶν ἐκ θέσφατα ᾔδη)
When the Lacedaimonians were overcome by famine, they consulted an oracle about a solution. The god told them to placate the deities of the Teucrians (this is what the people of Troy were called previously). [...]
Translation by R. Scott Smith et al.
The scholia says “this is what the people of Troy were called previously” probably because, in the myth, even the term “Trojan” comes from a king named Tros. Tros was the inspiration for the name Troy, which is why the inhabitants were called Trojans. In the same way, mythology tries to explain the name “Dardanus” with the character Dardanus and the name “Illium” with the character Ilus. The name “Mount Ida” maybe was related to the nymph Idaea, who had Teucer with the river Scamander, or even Idaeus, one of the sons of Dardanus.
Some sources mention characters that are considered eponymous, but I’ll only present Library as an example because it brings all three of them together:
[...] That country was ruled by a king, Teucer, son of the river Scamander and of a nymph Idaea, and the inhabitants of the country were called Teucrians after Teucer. Being welcomed by the king, and having received a share of the land and the king's daughter Batia, he built a city Dardanus, and when Teucer died he called the whole country Dardania.
And he had sons born to him, Ilus and Erichthonius, of whom Ilus died childless, and Erichthonius succeeded to the kingdom and marrying Astyoche, daughter of Simoeis, begat Tros. On succeeding to the kingdom, Tros called the country Troy after himself [...]
[...] there Ilus built a city and called it Ilium. [...]
Pseudo-Apollodorus’ Library, 3.12.1-3. Translation by J.G. Frazer.
In the case of Mount Ida, here is an example with Idaeus:
[...] Idaeus, the son of Dardanus, with part of the company occupied the mountains which are now called after him the Idaean mountains, and there built a temple to the Mother of the Gods and instituted mysteries and ceremonies which are observed to this day throughout all Phrygia. And Dardanus built a city named after himself in the region now called the Troad; the land was given to him by Teucer, the king, after whom the country was anciently called Teucris.
Dionysus Helicarnassus’ Roman Antiquities. Translation by Earnest Cary and Edward Spelman.
Establishing here the idea that these local names were attributed as homage to these mythological characters, it’s interesting that the scholia says “because he had been born of a Trojan woman, they called the child “Teúcros,” moving the accent back to create a proper noun. For the Trojans are called Teucroí after one of their rulers, Teucros”. The scholia doesn’t say “the name was in honor of the king”, the scholia says that the inspiration was the word itself that designated the Trojan people. In other words, Teucer was basically named after not a specific man, but his mother’s people as a whole, who were once named after that man. And the scholia doesn't say "Hesione named him", the scholia says "they". So this was probably a decision by more than one person. Telamon and Hesione, probably. I can see Hesione naming her son this because she wants to remember her home, after all she's not there because she wants to be (she's a captive), but Telamon? It's interesting that the apparent scholia indicates that Telamon agreed. I doubt he did it out of kindness, Telamon is usually consistently portrayed as a character with a bad temper (for example, see Apollonius Rhodius' Argonautica). Could it be that, in this case, Telamon wanted to sinalize Teucer as a half-Trojan boy? Not out of kindness to Hesione, but as a way of accentuating Teucer as a bastard of foreign blood? After all, he did indeed have favoritism towards Ajax, this is also consistent in the myth (see, for example, how he banished Teucer from Salamis because he blamed him for Ajax's death. In Sophocles' Ajax this element is emphasized). Also, a character called the equivalent of "Trojan" helping to destroy Troy is something… to think about. The possibility that Teucer was named after the people rather than the man makes it all the more interesting.
Anyway, I think this is headcanon. Of course, it seems quite purposeful that he is given a Trojan name as far as the narrative is concerned, but I honestly don't think they were thinking about specific details like Hesione and Telamon's inner motivations for the name when this was written. But I like my new headcanon!