Famous demi-gods were sometimes turned into gods. Namely:
But were there any mortals that were turned into gods, or immortal?
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Some members of this list count as gods, although none are, to my knowledge, regarded as true Olympians. While all gained immortality, I'm only counting a select few as gods because they were either venerated by a large group of people or recognized by other immortals as gods.
Glaucus: Originally a fisherman, he ate a "divine herb" from Cronos and became immortal. He helped the Argonauts on their quest, and for this he was venerated by fisherman and sailors, and accepted by other marine deities as a god of the sea. Philostratus the Elder writes
The story is that he once dwelt in ancient Anthedon and that he ate a certain grass on the seashore, and that when a wave came upon him unawares he was borne away to the haunts of the fishes. Now he is probably uttering some great oracle, for he excels in this art.
Glaucus is somewhat unusual (as the other examples will show) in that he became immortal purely by his own doing. Other figures gained immortality thanks to the favor of a god.
Orithyia: Originally a mortal, she was abducted and raped by the north wind, Boreas, who made her his wife. Two of their children, Calais and Zetes, sailed as Argonauts. Orithyia was venerated by the Athenians as a wind goddess.
Ino (known as the goddess Leucothea): Ino was, like Orithyia, once a mortal princess. She married Athamos (also mortal). Different versions of her life and death exist, but many end with her throwing herself and one of her children into the sea, possibly to escape the rage of Hera, because Ino had raised Dionysus as a boy. The sea deities (sometimes including Poseidon) then accepted her, like Glaucus, as a god, and she received the name of Leucothea. Her son, Melicertes, also became immortal, known as Palaemon, another god of the sea.
Bolina: A slightly more mundane example, Bolina was loved by Apollon. Fleeing him, she, like Ino, dove into the sea, and was given immortality, as a nymph (which for our purposes may count as a deity, although I don't believe she was venerated).
Since your examples are Greek, I presume your focus is exclusively on that mythology.
Dionysus and his maternal relatives, such as his mother, aunts, uncles, cousins and his grandparents Cadmus and Harmonia (as well as his great-uncle Iasion), have a mysterious and ambiguous relationship with immortality (and mortality). Almost all these characters acquire immortality (of some sort) at some point through some especially unorthodox means in their adventures.
Dionysus' mother Semele becomes the goddess Thyone, Semele's sister Ino becomes the goddess Leucothea and Ino's son Melicertes becomes the god Palaemon.
Semele's and Ino's sister Autonoe eventually attains immortality after having been married to the minor deity Aristaeus, who himself seems to originally have been a mortal like his brothers Idmon and Autuchus.
Harmonia, the mother of Semele, Ino and Autonoe, is the daughter of Ares and Aphrodite but seems to somehow start of as a mortal being raised by the Pleiad Electra, a lover of Zeus, together with Iasion, a son of Zeus and Electra. Iasion himself sort of becomes a minor god after being killed by his own father.
It's kind of difficult to tell if Plutus and Philomelus, Iasion's sons by Demeter, are considered immortal from birth or if they only gradually become this way similarly to their half-brother Dionysus.
Dionysus made his second cousin Ariadne into a goddess either before or after she killed herself on Naxos Island, whereupon he married her.
For full details on these and many others in general, check out Aaron Atsma's Complete List of Deified Men and Women in Greek Mythology, whom he calls the Apotheothenai.