Yes, there are several specifically named characters in this role.
Aaron Atsma's website The Theoi Project has a page thereon entitled Elysion [Elysium], detailing the geography (or, actually, different geographies according to the variety of writers from diverse periods in ancient Greek mythography) and citizenship of the Mythic Realm (he has a whole series of pages on the different "Mythic Realms" of Greek cosmology) in question.
One of the most mysterious characters of the mythology is a certain Lykos [Lycus]. He was the son of Poseidon by Kelaino [Celaeno], one of the seven Pleiades. The only other thing that we know about him is what Apollodorus tells us when he says in the Bibliotheka that Lykos' own father "Poseidon settled" him "in the Islands of the Blessed." It makes it sounds as though Lykos was still alive when this happened but the reference is brief and tantalisingly vague, mentioning nothing about any heroic adventures or anything else to warrant this special treatment.
(My only guess is that Lykos' parentage is quite similar to that of Hermes, who is rewarded for his extraordinary feat of thievery on the day of his birth by being made into an Olympian god. Lykos is Hermes' cousin both on his father's and mother's sides of the family, Hermes being the son of Zeus by another Pleiad. Other than that there's the probably unhelpful detail that Lykos means "Wolf," which need not be an indicator of anything particularly wolfish about the dude himself. The Roman writer Hyginus conflates this Lykos with the brother of King Nykteus [Nycteus] of Thebes.)
In the Epitome, Apollodorus tells us that after his death in the Trojan War, Akhilleus [Achilles] was married to the immortal Colchian princess witch Medeia [Medea]. In Book 4 of Apollonius Rhodius' Argonautika, while Akhilleus is still a baby but Medeia is having her adventures with Iason [Jason] and the rest of the Argonauts, Hera prophesies to Akhilleus' mother Thetis that this union will eventually take place.
A writer called Ptolemy Hephaestion, whose collection of alternate myths contains some fascinating story versions and anecdotes, says, however, that on the Islands of the Blessed, Akhilleus was united to Helen, the reputedly most beautiful woman in the world, over whom the Trojan War was waged. They even had a son there on the islands. This child of theirs was called Euphorion, named "after the fertility of this land", and he had wings!
There are other characters listed by Atsma as having become inhabitants of the Blessed Islands:
- Kadmos [Cadmus] and Harmonia
- Rhadamanthys; and Alkmene [Alcmene] the mother of Herakles [Heracles]
- Orpheus and Eurydike [Eurydice]
- Akhilleus' cousin Aias [Ajax], Akhilleus' son Neoptolemos, and
Akhilleus' relative Patroklos [Patroclus]
- Aias son of Oileus
- Odysseus' wife Penelope, and their son Telemakhos [Telemachus]
- Antilokhos [Antilochus] son of Nestor
- Diomedes, Iphigeneia, Memnon and Menelaus
Atsma says that his "list is far from exhaustive--any hero who possessed a hero-cult in ancient Greece was presumed to be transferred to Elysion." The "Elysion" page, typically, has a plethora of quotes from various ancient authors, including Homer, Hesiod, Pindar, Euripides, Aeschylus, Strabo, Diodorus Siculus, etc, several Romans making up part of the list as well. See the linked page for more detail.