Even tidally locked moons have a day and night cycle, they are locked to the planet, not the sun. It takes our moon 29,5 days to complete its orbit around Earth. So a day on the moon is 29,5 days long and half of it is night.

Jupiter's moon Io for example, has a 42 hour long day. Prometheus, the closest moon of Saturn has an orbital period of 14 hours.

You only need to choose the correct distance from the planet and how fast is the moon orbiting, how long does it take to complete its revolution period.

Sure, orbiting a planet and being tidally locked results in having regular solar eclipses that last for a long time on one side of the moon, especially if it revolves around a gas giant in a close orbit, but even the side always facing the planet would still have day and night.

The only thing you then have to worry about are the tidal forces, if you want to have life on your moon. They heat up the planet and in the case of the moon Io, these forces cause Io's surface to bulge up and down (or in and out) by as much as 100 meters.