My interpretations of the titles are:
1. Leviathan Wakes
The Leviathan is a sea monster in Judaism, most well known from its appearance in the Book of Job.
The title refers to the waking of the long-dormant protomolecule from Phoebe, moon of Saturn. At the beginning of the series the protomolecule is mostly treated as a terrifying alien threat, an incurable disease that turns humans into "vomit zombies" and breaks the laws of physics as it somehow flies the asteroid Eros until it crashes onto Venus.
2. Caliban's War
Caliban is a half human, half monster character from Shakespeare's play The Tempest.
The title refers to the project which created hybrid human-protomolecule soldiers and the war fought with and over them.
3. Abaddon's Gate
Abaddon is a Hebrew name for the abyss or pit of destruction, as well as an angel personifying the same.
The title refers to the characters passing through the protomolecule's ring gate and the rapid forced braking they experience and the destruction that follows.
4. Cibola Burn
The Seven Cities of Cíbola (or Gold) were mythical cities of the Americas filled with gold and other riches, which the Spanish colonists search in vain for. The Spanish never found cities of literal gold, but they did find many resources to extract, as well as the ruins of much older civilisations.
The title refers to Ilus, a planet rich in lithium, and the mining rush that follows. The planet is settled by both Belter refugees and a mining company from Earth who are scrambling to claim all the resources they can get their hands on. Once they get there, they also discover technology from the Ring Builders, waiting to be turned on, and exploited. Like in a gold rush, colonists burn hard (accelerate heavily) to reach Ilus and other planets before other colonists can compete with them.
5. Nemesis Games
Nemesis is a Greek goddess who enacts retribution against the arrogant.
The title refers to Marco Inaros and his Belter "Free Navy" who attack the Earth for what they perceive to be centuries of arrogant oppression of the Belt. But despite Marco's claim to represent all the Belt, he cares little for them, and being seen as victorious matters more than actually coming out ahead. In actuality, Marco reacts on the fly, cuts his losses, and lies that it was his plan all along, which I think can fairly be said to be treating war like a game.
6. Babylon's Ashes
Babylon was the capital city of the Babylonian Empire. In Jewish and Christian symbolism it stood for the corrupt world, evil, and eventually the Roman Empire.
After the rocks fell Earth all but completely collapses, while Mars becomes impotent after a third of its navy mutinies and leaves the solar system. The Belt likewise falls into squabbling factions as the Marco's Free Navy disrupts the unity of the OPA. So the title depicts the old political order of the solar system as the empire of Babylon burned down to ashes, while foreshadowing the next book's title and the new empire that will rise from those ashes.
7. Persepolis Rising
Persepolis was the capital of the Persian Empire.
The title refers to the re-emergence of the hidden Laconian Empire which, just as the Persian Empire conquered the Babylonian Empire, then conquered the "Babylon" of Earth, Mars, and all the other inhabited systems. The Laconians saw themselves as the continuation of the dream of Mars, after Mars's own dream of terraforming became pointless when the gates opened, and believed they had the right and duty to rule over all the systems as the centre of the new human empire.
8. Tiamat's Wrath
Tiamat is the Babylonian goddess of the sea and chaos.
The title refers the unknown alien race which had previously defeated the protomolecule's Ring Builders race and is now attacking humanity (or defending itself), by a variety of means, including swallowing ships as they go through the ring gates, deleting physical matter from ships and humans alike, and causing mysterious system-wide consciousness breaks. The motivations of this alien race are unknowable as the sea, if indeed it is even right to call them an alien race.
9. Leviathan Falls
I think this title mostly serves to bookend the series, completing the story begun in Leviathan Wakes. But it also refers to the end of the Ring Builders' protomolecule-powered dream, the rejection of their hivemind, and the end of the ring network. Instead of inheriting the protomolecule technology and civilisation, humanity will have to find its own way back to the stars.