TOI-1136 and Kepler-90 (and Hyperion) suggest that, if the gravity isn't too bad, one could squeeze a 4:3:2 stable-orbit space station between binary-resonant planets. Like: between Gliese 1002 b and c, if you don't mind the 15.8ly walk. Or: between Mimas and Tethys, or Enceladus and Dione, or pretty-much anywhere around Uranus. Let's talk betwixt Europa and Ganymede.
We should be able to find "Hill Sphere" or "Hill Radius" +Ganymede on the internet like Bagenal and Dols offer for Io but feh. So I'll do it: wedge this between Ganymede 1070400-31717 = 1,038,683 km; and Europa 670900+13652 = 684,552 km, from Jupiter's barycentre, by calculation. That's 300,000 km of space between; plenty of room for a Hyperion-sized O'Neill with rubble-shield.
Between 3.551181 days and 7.15455296 days, 4:3:2 suggests 4.7349-4.7697 days. What's fifty minutes between friends? Laplace has a better equation for 4:2:1; papers exist also for Kepler-90def. But this is a blog, not a physics seminar. 4.75 days it is (UPDATE 12/24 almost!). With a Keplerian mass 23959722339086384: semimajor is 814622 which I'll round to 814600 814216 km, 30° behind conjunction.
There's nothing there now which I must ascribe to the tidal-forces of Jupiter and Io. Hence the bounds around the period and meanmotion! But the stationkeeping shouldn't be godawful, within the limits of docking craft and launching them.
What would we get, playing middleman (well, 42%-man, lagging) here? At the very least here's the rescue-station for inadequately-propelled shuttles between Ganymede (which I shield) and Europa (which Twitter people tell me is shielded, but I'm not betting on it). By "inadequately-propelled" we can start with Hohmann; and we're told of even less delta-V options, on account Jupiter's is a multibody system, not just three-body.
For la Maison de Pierre-Simon other options include communication-relays. 4:2:1 systems mean that one or the other might be at Jove opposition at the same time the other two are in conjunction. And - since these moons are rotationally-locked - the Jove-facing side of an inner moon will not be able to talk to the outer moon(s) at that conjunction. Maybe the relay is not light-focusing at first but as noted, we got a lotta space-between.
Politics politics politics! This is the spot for Europa/Ganymede diplomacy. Also for high-G training if these two moons need a coördinated response against Callisto and beyond.
As for keeping the lights on: Some tidal heating might help keep the inner parts physically warmer, like under Europa. The ionic wind can work for additional energy-generation; both will supplement the Sun, weak out here.
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