The gas giant can only eclipse the sun when the moon is on the far side of the gas giant from the sun. The eclipse is then seen from everywhere on the day side of the moon. I'm using the natural definition of "day", relative to the sun, rather than the gas giant.
Now, that doesn't happen exactly once per orbit of the moon around the gas giant. During the time the moon takes to orbit the gas giant, the gas giant moves a bit in its orbit, so the moon has to make slightly more than one orbit of the gas giant to reach the next eclipse.
So if P is the orbital period of the gas giant around the sun, and p is the orbital period of the moon around the (edit) gas giant, the correct rotation period for the moon to have an eclipse exactly every day is $p * (1+(p/P))$. The eclipse happens at the same time each day, and only one side of the moon ever sees the eclipse.
I've assumed that the planetary system works in what we think of as the normal way, like our own solar system, with all the orbits and rotations in the same direction.
For any plausible setup, with P much larger than p, the moon is rotating a little slower than tide-locked, and that raises the question of how it got to be that way. As best we know, moons would start off rotating rapidly and gradually spin down under tidal forces until they tide-lock. Some external force would be required to slow the spin further, such as a collision.
Addendum: If the moon is orbiting in the opposite direction to all the other circular paths, then its rotation period needs to be a little less than its orbital period to have an eclipse happen every day: $p * (1-(p/P))$.
Addendum 2: To have an eclipse clearly visible from everywhere on the moon every moon-day, you need to have the moon rotating much slower than tide-locked, so that it completes several orbits per day. You probably want at least three orbits per day, to ensure that the sun is definitely above the horizon for at least one of them. This means that some places will see at least part of more than one eclipse per day, but this is unavoidable.
If it were me, I'd be tempted to have $pi$ orbits per day, which would tempt the inhabitants of the moon towards mystical numerology as a part of their early development of astronomy and astrology.
If you were trying to have realistic astrophysics, and a day length that isn't vastly longer than that of Earth, I'd be worried that the moon would have to orbit too close to the gas giant, and would be at risk of being inside the Roche Limit and falling apart. However, for an explicitly fantasy world, don't worry about this.