Can tidally locked planets have liquid oceans on their night side, and would such a planet be habitable? Let's say we have a tidally locked planet orbiting a star. And let's say that the conditions on its surface are just right for water to exist on its surface. Conventional wisdom says that the water on the day side will be liquid, while the night side will be covered with a sheet of ice.

However, I'm wondering if this is a hard and fast rule. What I want to know is if it is possible for tidally locked worlds to have liquid water persist on its night side, and what conditions could bring this about. And would those conditions still leave the world habitable?
Short Answer:

With current knowledge nobody knows.  Some calculations suggest that it might be possible for a tidally locked planet to be habitable and to have life.

Long Answer:

The first stop searching for information about the habitability of tidally locked planets in red dwarf star systems might be the Wikipedia article Planetary Habitability.


  Astronomers for many years ruled out red dwarfs as potential abodes for life. Their small size (from 0.08 to 0.45 solar masses) means that their nuclear reactions proceed exceptionally slowly, and they emit very little light (from 3% of that produced by the Sun to as little as 0.01%). Any planet in orbit around a red dwarf would have to huddle very close to its parent star to attain Earth-like surface temperatures; from 0.3 AU (just inside the orbit of Mercury) for a star like Lacaille 8760, to as little as 0.032 AU for a star like Proxima Centauri[76] (such a world would have a year lasting just 6.3 days). At those distances, the star's gravity would cause tidal locking. One side of the planet would eternally face the star, while the other would always face away from it. The only ways in which potential life could avoid either an inferno or a deep freeze would be if the planet had an atmosphere thick enough to transfer the star's heat from the day side to the night side, or if there was a gas giant in the habitable zone, with a habitable moon, which would be locked to the planet instead of the star, allowing a more even distribution of radiation over the planet. It was long assumed that such a thick atmosphere would prevent sunlight from reaching the surface in the first place, preventing photosynthesis.
  
  This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibars (0.10 atm), for the star's heat to be effectively carried to the night side.[77] This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.[78]


https://en.wikipedia.org/wiki/Planetary_habitability#Red_dwarf_systems1

This article links to a longer discussion of the habitability of red dwarf star systems.

https://en.wikipedia.org/wiki/Habitability_of_red_dwarf_systems2

And of course the various sources listed in those articles are worth studying.