Why does my RTC module keep dropping out when I touch the wires?

Your instinct about the jumpers is right. I ran an outdoor temp logger on a DS3231 for a couple years and it did exactly the same thing, worked fine then dropped off the bus whenever the box got bumped. Those cheap duponts lose thier spring tension fast, and once the socket can't grip the pin properly you get intermittent contact that looks exactly like what you're describing. If they slide on with basially no resistance, that's your problem right there. For a permanent install like an aquarium timer, my pick would be to ditch the jumpers entirely. Either solder the four wires straight to the pins, or better yet use a 4-pin JST connector if you want to keep it serviceable. JST-XH has a proper clip that clicks and doesn't work loose from vibration. If you've got a crimp tool you can make your own dupont ends with decent ferrules too, but honestly for four wires direct soldering is simpler and bulletproof. On the loose headers, don't ignore those. If the pin wobbles in the plastic body, the internal connection to the solder joint on the board can crack over time, and no amount of good wiring downstream will fix it. A quick touch with a soldering iron to reflow each pin takes two minutes and is cheap insurance. If the pin is loose in the plastic housing itself, that header is worn out and worth replacing rather than trusting.

Hot glue does work fine as a strain relief if you end up keeping jumpers. Put a small dab on the plastic body of the connector and a bit onto whatever it's sitting next to, and it peels off later with a fingernail without much mess. Just don't flood the whole area or you'll never get it apart cleanly.

One other thing worth checking while you're at it: make sure the grounds are solid between the Arduino and the module. A flaky ground on an I2C bus causes dropouts that look exactly like this even when the data lines are fine. And keep those two data wires short, a few inches is plenty, long jumpers on SDA and SLC pick up noise and make marginal connections fail more often than they should.

One more thing to check before you blame the wires entirely. I chased a similar dropout on mine for weeks, swapped every jumper I owned, and it turned out the header solder pads on the module itself had a hairline crack. Gave the board a gentle flex test while it was running and the sketch lost it instantly. A 30 second touch-up with the iron and it's been solid ever since.

Also, if you do end up soldering direct, leave yourself a couple inches of slack wire so you can pull the module out for battery changes without desoldering anything. Ask me how I know.

Another angle on this since it's an aquarium build: keep the module out of the damp air. I had a controller near a fish tank years back and the humidity crept into everything, pins tarnished and contacts went dodgy even though the wiring was solid. Put the electronics in a sealed-ish box and just bring the wires out through a grommet, and give the joints a look every few months for green crust.

Also, once you've got the final wiring sorted, do what we did before fancy crimp tools existed: tug test every lead gently while the sketch is running. If the clock drops off the serial monitor during a light pull, it's not done yet.

Update: tried that flex test someone suggested while it was running and sure enough the sketch lost the module when I pressed near one corner of the board. So it might not be the jumpers after all. Can't see any crack on the solder pads with the naked eye though. Would a magnifying glass be enough to spot a hairline, or should I just reflow all four pins and hope for the best?