How do the day/night temperature variations of moons compare to those of their planets? Does the planet's eclipse have a significant impact on the flux of light hitting the moon? Does tidal locking have any effect on the day-night difference of the planet?
 A: These questions are only lightly related, but I'll try to answer them all.
As discussed in another question (will linkify if I can find it), the atmosphere of a spherical body is more important than raw distance from the Sun- Venus' day and night are both hotter than Mercury's day, and Mercury's night is frigid. Also, all of the gas giants except Jupiter (I think) actually give off more energy than they receive from the Sun. [EDIT: Oh yeah, I forgot to mention: e.g., Io fries while Callisto is frozen.] So, apart from a general correlation between distance and temperature, they are not strongly related.
Moon brightness is tricky! Consider how bright it is on Earth during a full Moon. Now consider  that Earth will appear to have about 16 times as much area as seen from the Moon than vice versa, and the Earth has an albedo about three times higher, so "Full Earth" provides about 50 times as much light as a full Moon. http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
Now, if we turn our attention to a gas giant, the situation will be even more extreme! In fact, my PhD adviser actually did the calculations, and for the planet-facing side of a tidally locked moon of a gas giant, the brightest time is during a total eclipse of the Sun, because there is a "Full Jupiter" (or other gas giant) out! The darkest time would actually be during the crescent phase of the parent planet, I think. For non-tidally locked moons, you'd have to run the calculations individually.
Are you really asking about how tidal locking of the moon affects the day/night difference on the planet? Assuming so, remember that the planet keeps rotating and revolving, while the moon still revolves but does not rotate (in the reference frame of the planet!). With tidal locking, the moon is guaranteed to present a very nearly constant "performance" for the planet. If the moon were *un*locked yet presented very different albedo on one side compared to the other, you could have substantial differences in apparent brightness. The problem, though, is that a moon with very different faces is exactly the kind most likely to be locked, because it gives the tidal forces the best "grip".
