I've tried really hard with my "planet simulator" to get inhabitable moons. It has never coughed up a single one in around a million trials.

The reasons:-

1) The moons tend to be too small.

(I scale moon size with gas giant size, just as I scale planet size with star size - it seems "reasonable". I also scale moon *number* with gas giant size - again reasonable, but this limits the size of the largest moon to around the square or cube root of the gas giant. Since gas giants more than 13 Mjup experience deuterium fusion, this is the limit on habitability and it's almost always too small to retain an atmosphere when close enough to the sun for liquid water - e.g. Titan works where it is, but wouldn't here)

2) The moons tend to be too wet.

(all the evidence from our solar system and from models for formation of gas giants has them forming in ice-dominated regions of the stellar nebular. This leads inevitably to ice-rich compositions, unless you roast it through tidal flexing, like Io. A warm icy moon becomes a waterworld wit a very deep ocean. Habitable yes, but not by technological, land-based, star-gazing civilisations - ever tried inventing radio underwater?)

of course, I could arbitrarily change the algorithms to make such moons frequent outcomes, but that's flying in the face of the (limited) evidence.

Nonetheless, such a world would be very interesting with a "day" measured by the orbit of the (tide-locked) moon round its primary, and a superimposed longer-term climate cycle due to the eccentricity of the gas giant's orbit.

One side would permanently face away from the gas giant, seing a strong day/night cycle. The other would permanently face the primary and have a reduced cycle due to eclipsing every "noon" for 10 - 30 degrees of orbit and milder nights due to IR glow from the primary.

You could even partially escape the tide-locking by having a 3:2 resonant orbit like Mercury.

Some good SF scenarios in there, but I repeat that they are hard to substantiate.

Then again, nobody expected "hot Jupiters"...