For the Earth-Moon system, the orbit of the Moon is at a slight incline compared to the plane of the ecliptic. This incline is enough for there to be eclipses roughly twice a year rather than every lunar month.
But Jupiter is much bigger, so its shadow is bigger also. So it seems more likely for a satellite to go into its shadow. How often does this happen?
Every Galilean moon and inner moon go into lunar eclipse once per orbit.
How often is there a lunar eclipse of the Jovian moons?
I set up a little animation of the Galilean moons (I didn't include the inner moons), and made a few assumptions (negligible moon size, sun at infinity, perfectly circular orbits) and found that there is a lunar eclipse in the Jovian system just under 10% of the time.
Here's a bigger version of the second graph.
The orbits of the Galilean satellites have a roughly 2° incline.
Based on their distance from Jupiter and on the radius of Jupiter, I computed the apparent diameter of Jupiter from these satellites. The apparent diameter of Jupiter from each of these satellites is 19°, 12°, 7.5° and 4.3° for Io, Europa, Ganymede and Callisto respectively.
Therefore, the Galilean satellites pass in the shadow of Jupiter once per orbit around Jupiter.
For satellites with a more inclined plane like Themisto and Himalia, eclipses will happen less often, perhaps every six years (half of a Jupiter revolution).
NOTE: This (non)-answer is wrong, because it only computes when an Io-centric viewer sees at least a partial solar eclipse:
If there is a partial solar eclipse at Io's center, there is necessarily at least a partial solar eclipse somewhere on Io's surface...
However, a partial solar eclipse on Io's surface just means that part of Io is slightly darker, not totally dark, so not sure if that counts as a lunar eclipse from Jupiter...
According to https://www.merriam-webster.com/dictionary/lunar%20eclipse a lunar eclipse is defined as "an eclipse in which the full moon passes partially or wholly through the umbra of the earth's shadow". Thus, there must be a total solar eclipse somewhere on the moon-- partial isn't good enough.
It's possible to have a partial or even total eclipse on Io's surface but no eclipse at all at Io's center.
I ran my code for Earth solar eclipses, and everything it returned was an eclipse, but it didn't return all eclipses (for the reason above).
I thought I was doing something wrong, but https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/C/cspice/gfoclt_c.html#Examples (the documentation for the function I'm using) gives the example "Find occultations of the Sun by the Moon (that is, solar eclipses) as seen from the center of the Earth over the month December, 2001" (emphasis added).
Apparently, finding total and partial eclipses on the surface of Earth is much more difficult. I'm trying an approach that computes the umbral and penumbral cones and intersects them with spheres, but I haven't gotten very far.
This not an answer, but, if you're extremely interested, you can use https://wgc.jpl.nasa.gov:8443/webgeocalc/#OccultationFinder with these parameters:
and others (you'll need to fill in other boxes, but that's fairly basic), you'll see when Jupiter blocks the Sun as viewed from Io, which means the Jovians will see an Io-nian lunar eclipse. You can use similar techniques for the other moons and/or use the CSPICE libraries (see http://astronomy.stackexchange.com/questions/13488/) to make your own calculations.
NOTE: for observer, make sure to use Io the moon, not Io the asteroid. The one with a number in front of it is the asteroid.
Of course, you could also use a planetarium program (like Stellarium) that lets you view the sky from Jupiter's moons (or even Jupiter itself-- I'm not sure Stellarium computes lunar eclipses, but it should)
One final thought: in theory, one Jovian moon could eclipse the Sun as viewed from another Jovian moon -- not sure if that counts as a lunar eclipse (can't happen here on Earth since we have only one moon).
NOTE: Of course, this question can be extended to "when seen from X, how often does Y appear to be obscured, even partially, by any third object Z". Simple observer Earth examples:
solar eclipse: X = Earth, Y = Sun, Z = Moon
lunar eclipse: X = Moon, Y = Sun, Z = Earth
Venus transit: X = Earth, Y = Sun, Z = Venus
Note for a lunar eclipse, we say that observers on the Moon see the Earth eclipse the sun, darkening (portions of) the moon, which means Earth viewers see Earth's shadow on the Moon.
This should be solvable by running through all the combinations in CSPICE, but it might be useful to find "interesting" obscurations such as where X, Y, or Z is a planet and/or a large planetary moon.
