What you've got is an EMC (Electromagnetic Compatibility) problem.
The computer sounds like one of the older, non-coded variety, which are prone to being very easily susceptible to interference from other source. Typically this is in an LED front light, where the LED illumination is modulated to control it's brightness, and power consumption.
Equally, it sounds like from comments made by others, that this particular Etrex is also poorly shielded, so the CPU is managing to radiate enough RF energy to interfere with the computer.
Generally this doesn't tend to happen, since there is normally sufficient shielding on any microcontroller (metal case), and they run on such low power levels, and often at relatively low speeds, that EMC tends not to rear it's ugly head.
Unfortunately, the wireless computers use very low levels of signal, to make the battery last as long as possible, so even very slight levels of interference can stop them from working.
Some modern designs of bicycle computer use coding schemes, which make them more immune to interference, but also tends to make them more expensive, they are becoming more common though, so the price is also dropping.
With the equipment you've got, you really only have two solutions, and only one is likely to be practical.
(i) Increase the shielding on the Etrex. Essentially you need to put more metal around the Etrex to reduce the RF energy that's radiated by it. This may be difficult to do in a way that is rugged enough, and still allows you to see the screen, and mount the GPS. You can't shield the computer, because it needs to see the RF energy from the transmitter. There are other techniques, but they're probably not practical for people on here. You may be able to get away with removing the case from the Etrex, painting the insides with conductive paint, and then reassembling it. The trouble with that, is that the paint isn't easily available, it's expensive, the box reassembly will likely be interfered with, and it's harder to deal with the screen, which is also a damned big (RF) hole in the case.
(ii) As others have said, increase the separation between the devices. If they are currently close together, then even a small increase could be sufficient. Even something like swapping them over may help, if the problematic internal components happen to be on the right (or wrong!) sides. Of course with that specific scenario, you could also make things worse!
