I've noticed that a number of creatures are related to the Greek deities. By the oldest account, Echidna, the mother of monsters, is the daughter of sea gods. This accounts for her own kids being related to deities. But what about the monsters that aren't children of Echidna? Are they related to other gods?
Almost all of the monsters of Greek myth are clearly said to have been children of deities or at least were quite closely related to the gods, whether the divinities in view were Titanic, primaeval or otherwise.
Good Old Regular Wildlife
There is also an assortment of monstrous beasts, quite popular especially in later mythography, which appear to have been thought of simply as naturally occurring animals like any other ordinary critters such as dogs, cattle, birds or tigers, etc.
Typically these "animals" were understood to dwell quite far outside Greece and the Roman Empire, at the edges of the world, such as Greco-Roman writers believed Africa, Arabia and India were located at. It was often essentially a matter of most of their readership never having been to these places and thus all manner of tall tales grew up about the fabulous populations thereof.
Aelian's book On Animals catalogues a good number of these, such as:
- Neades, which were monsters, inhabiting the island of Samos in prehistoric times, whose powerful roars could split the ground;
- the Scolopendra (not to be confused with the modern scientific term for a genus of centipedes), which he says was the largest of all sea monsters;
- various Indian sea monsters which he describes as fish-like things that had the heads of lions, leopards, sheep and wolves;
- unicorns, also originating in India, some of which were actually single-horned donkeys the size of horses, with some interesting colour features as well as other unusual characteristics which made them very dangerous;
- gigantic Indian snakes which preyed on elephants (Philostratus tells us in his Life of Apollonius of Tyana that griffins, another naturally occurring beast, had an advantage over these snakes because, unlike the serpents, they had wings and therefore could fly); and
- gigantic Aethiopian snakes dwelling in the same land as the virtuous long-lived (or immortal) Africans among whom the Greek gods used to take their annual vacations such as Homer's Iliad describes (these serpents being the largest such reptiles, says Aelian, they grew to as long as 180 feet [about 55 metres = at least seven times as long as some of the biggest anacondas] and, also living longer than any other animal, they too, like the Indian variety thereof, preyed upon elephants).
The ancients believed they were the adult-form of the small fish we call the "sea-horse". Hippokampoi were the mounts of Nereid nymphs and sea-gods, and Poseidon drove a chariot drawn by two or four of the creatures.
Other fish-tailed land animals which appear in ancient art include the "Leokampos" (fish-tailed lion), "Taurokampos" (fish-tailed bull), "Pardalokampos" (fish-tailed leopard), and "Aigikampos" (fish-tailed goat). The last was the form of the constellation Capricorn. Fabulous creatures of this type were believed to be common in the Indian Ocean.
The parentage of a few of the more classical monsters is not mentioned, such as the sea monster which Poseidon sent to Aethiopia and which Perseus rescued the local princess Andromeda from being sacrificed to.
Most of the rest, however, usually have at least one deity for a parent. The clearest exception that I can think of is the example of the grizzly man-eating giants Agrius and Oreius, whose mother Polyphonte had conceived them by mating with a bear. Although neither of these giants' parents was a deity, Polyphonte's mother Thrassa was the daughter of the war-god Ares, thus making Agrius and Oreius the great-grandsons of Ares.
The parentage of Agrius and Oreius is quite similar to that of the Minotaur, whose father was a bull and whose mother Pasiphae was queen of Crete Island. Since Pasiphae was a goddess, this technically made the Minotaur part god and part ox. The Minotaur's grandfather was Helios, the Sun.
The hundred-eyed giant Argus Panoptes, "Argus All-Eyes," has a variety of different parentages ascribed to him, most having to do with the royalty of the city of Argos. Several authors say that Argus, like many other giants, was an Autochthon, which is to say that he was born spontaneously out of the ground so that Gaia, the Earth, was called his mother.
One alternative says that Argus was a son of the River Inachus, who was also the father of the Naiad Io. After Zeus had almost been caught in the act of cheating on Hera with Io but then quickly disguised Io as a cow and pretended that that's all she was, Hera took the cow and recruited Argus Panoptes to guard her. If Argus's father was Inachus, this would mean that Argus was guarding his own sister.
Apollodorus lists a few other versions of Argus's origin, in one of which his mother Ismene is a Naiad of the Asopus River, and his father is also called Argus. This other Argus, after whom the city of Argos was named, was a grandson of Phoroneus, who in turn was yet another son of Inachus. In an additional version, the mother of the All-Eyed Giant is the Naiad Mycene, a daughter of Inachus.
In every single other example of monster that comes to mind, at least one of the creature's parents is a major deity, mostly either Gaia or Poseidon or both.
The Roman writer Hyginus tells of a primaeval Gorgon named Aega or Aex, whose father was the Sun. She had a dazzlingly beautiful body, "but in contrast to this beauty, had a most horrible face", and the Titans were so terrified of her, or so overwhelmed by her brilliance, that they begged their mother, the Earth, to hide her from them. Mother Earth complied by concealing Aega in a cave on Crete Island, where this solar Gorgon became a nurse of the infant Zeus.
When war broke out between Zeus and the Titans, an oracle told Zeus that if he wanted to win, he should wear armour made from a goat-skin combined with the head of Aega, who was presumably already dead at this point. Zeus made the prescribed ingredients into his famous shield, arm-guard or breastplate called the Aegis, which acquired its name from the Gorgon. Using this he was thus able to conquer his enemies, and it later became a signature emblem of Zeus's daughter Athena.
In the better-known classical story involving a Gorgon, there are three of them: triplet sisters named Medusa, Sthenno and Euryale, who were the daughters of the sea-god Phorcys by the sea-monster Ceto, who is both Phorcys's sister and his wife. This is from Hesiod's Theogony, the source that you reference in which Echidna is the offspring of a sea-god. Here, Echidna would be a sister of the Gorgons, who have another set of triplet sisters, the Graeae, "Grey Ones."
In the Theogony, Ladon, the sleepless hundred-headed who guards the garden of the Hesperides and is killed by Heracles (Hercules), is a brother of Echidna, and of the Gorgons and the Graeae. Apollodorus and Hyginus say, rather, that he is a son of Typhoeus and Echidna.
By a goddess named Crataeis (who was probably an Oceanid of the Underworld), Phorcys was the father of two beautiful sea-nymphs, Scylla and Thoosa. Out of jealously the immortal witch Circe, using a noxious concoction, changed Scylla into a fearsome monster encountered by Odysseus on his calamitous journey home from Troy. By Poseidon, Thoosa became the mother of the Cyclops Polyphemus, whom Odysseus also met under less than ideal circumstances.
According to Servius' commentary on the Aeneid, Gaia bore Poseidon a voracious gigantic daughter named Charybdis, who stole oxen from Heracles. With a lightning-blast, Zeus hurled her into the sea, where she transformed into the famous whirlpool, still called Charybdis, which was paired with Scylla as a danger to sailors. Apollodorus says that Gaia also bore Poseidon the Libyan giant Antaeus, whom Heracles wrestled with and killed.
The giant Laestrygon, from whom the monstrous Laestrygones of the Odyssey are descended, was also a son of Poseidon and Gaia, according Hesiod's Catalogues of Women. The Catalogues also mention a race of dog-headed people called the Hemikynoi, "Half-Dogs," whom later writers called the Cynocephali, "Dog-Heads." These were Gaia's offspring either by Poseidon or by Zeus' Egyptian son Epaphus.
The golden fleece which the Argonauts went to fetch from Colchis originally belonged to Chrysomallus, a golden, winged, talking ram which was the offspring of Poseidon by a princess named Theophane.
The famous winged horse Pegasus was a son of Poseidon and Medusa, and the triple-bodied red ogre Geryones, whose oxen Heracles had to steal in order to fulfill his tenth labour, was the grandson of Poseidon and Medusa.
According to Quintus Smyrnaeus' Fall of Troy, the four fire-breathing horses of the war-god Ares were the offspring of Boreas, the North Wind, by one of the Erinnyes.
Early in his career Apollo killed two gigantic offspring of Gaia. The first was the dragon Python, in one of whose birth stories this dragon hatched from the mud which had silted together after the Flood was over. The other offspring was the giant Tityus, who was actually a son of Zeus by a princess named Elara. Having debauched Elara, Zeus buried her under the earth for fear of his wife Hera's jealousy. Their son Tityus, a child of enormous size, burst forth from the ground some time afterwards, having been given a double birth, emerging first from Elara and then from Gaia's body. When Tityus had grown up, he tried to rape Apollo's mother Leto and paid for the offence by being shot to death by Apollo.
According to Pindar and Philostratus the Elder, the Centaurs of Thessaly were the offspring of Centaurus, a son of Apollo. In Apollodorus' Library, the Centaur Pholus is the son of the god Seilenus and the nymph Melia.
Hesiod says that the Titan Oceanus was the father of a pair twin monkey-like gnomes called the Cercopes, whom Heracles encountered in his adventures.
There's more where that came from in The Theoi Project's quite comprehensive Bestiary.