GPS mount resistant to vibration?

Perhaps a shim under the GPS device would make the mount hold the device better?

Just a piece of paper might work, or a thin piece of plastic.

I have exactly the same type of mount and the same type of GPS with it, and after a few years of use the contact interface has become worn out, just as you describe.

The problem was exacerbated by the fact that it was both the mount guides and the GPS unit's backplate grooves that got widened, so replacing just the mount part would not be very effective.

What I did was to use molding plastic that hardens over 24 hours. There are brands such as Sugru, Kintsu glue etc. that sell it.

Spread a very tiny amount of the substance along the rails of the mount, as well as inside the grooves of the device's backplate.

When hardened, the resulting silicon creates quite extra friction when sliding the device onto the mount. This makes the rattling go away, and secures the device.

Additionally I put some of the stuff onto the "tooth" that latches into the backplate (left side of your picture) and also the square "hole" in the middle of the mount so that it will take up some slack created by the wearing process.

The restoration has worked for me so far. I do not expect it to last forever however, but the procedure can be repeated later when the surfaces are worn again.

Even before I've encountered the problem with the worn-out mount plate, I've always used a piece of a thin cord to tie the GPS device to the handlebars. When I crash, zip ties that hold the mounting plate can snap if something strikes the handlebars (it did happen several times). The GPS can then fly away into an unknown direction if not for the cord.

The same cord is used to bear the GPS on the neck when I am off the bike (I once lost my GPS in the water when crossing/swimming across a wide smelly ditch in the middle of the night. I won't ever dare to hold a GPS device not secured by a cord).

We don't normally do product recommendations here, but it may be possible to address the issues in a general sense.

The eTrex 30 does not actually appear to be a bike-specific GPS, at least not in the way many cyclists understand the term. In the Garmin line, the entire Edge series are bike-specific GPSes, but their navigation capabilities are usually more limited. The eTrex appears to be a multipurpose GPS. It appears to me that it would usually be handheld, or put in a bracket mount in a vehicle. Garmin's site does say that it can be used in all terrain vehicles and on bicycles. The mount they describe as a bicycle mount is pictured below.

I assume the GPS has rails on its body that slide into the mount, and the mount zip ties to the bicycle. In contrast, the bike-specific GPS units use a twist mount. There's a mount molded into the computer's body. That interfaces with a corresponding puck on a more dedicated computer mount, e.g. a Garmin out-front mount pictured below. I think Garmin sometimes calls this mount a quarter-turn mount.

This style of mount, or the equivalents made by other manufacturers like Lezyne and Wahoo, are secure enough for mountain biking. GPS makers do often supply a stem mount secured with zip ties alone. Without having ever used an eTrex, I would guess that the rail mounting system is less secure than the bike-specific mounting system, assuming the OP used a genuine Garmin mount as pointed out by @ChrisH in comments. Maybe the rails suffice for users on smooth roads but not off-road. I have no personal experience, and these details weren't given in the question.

It is possible that there are more rugged third party mounts for bicycles. This is something the OP would need to investigate, as I don't keep track of this space. It's possible some specific communities may have come up with better mounting solutions, e.g. as discussed on this thread on bikepacking.com, or solutions that involve enhancing the existing mount as discussed in other answers.

Alternatives to the eTrex do exist. The Quad Lock mount system for smartphones, reviewed here by Cyclingtips, is actually quite secure and provides a phone case for major smartphone models. However, using your phone for navigation will drain the battery relatively fast, as the phone has to power the screen plus continuously search for GPS signals in addition to running the usual apps in the background. This may not be a problem on shorter rides in familiar areas. If the OP needs to do a lot of navigation, it is not ideal.

Garmin's more dedicated cycling computers have some options. I realize that OP prefers not to spend more money, but this option is worth mentioning because most bike-specific GPS mounts are pretty secure. The Edge Explore appears to be a touring-oriented GPS with a touchscreen and better navigation functions than, say, the Edge 130 or 530. These are more oriented to leisure or performance cyclists who either are on roads they know and are just tracking rides, or who are following a pre-provided course (e.g. given by a ride organizer, downloaded from a site like Ridewithgps.com). Their on-unit navigation functions are a lot more basic. I think the Edge 830 and 1030 have more advanced on-unit navigation functions as well. I am focusing on navigation capability because I assume the OP is looking for this because they referenced the eTrex in the original question. For alternative brands, I believe Lezyne and Bryton make good value for money GPS units, but I'm uncertain how extensive their navigation capabilities are.

If the OP knows their area well and just wants to track miles and ride leisurely, then actually, putting the eTrex in your back pocket would suffice. For that matter, starting Strava on your phone and putting that away will also suffice; this does raise battery consumption, but not excessively if you aren't continuously referring to the phone on the ride. I remember recording a full century ride on my old iPhone 4 and still having about 20% battery at the finish, although this will obviously depend on any other apps you have running in the background plus your battery health. Riding without any electronic input in this fashion can free you up to concentrate on the joys of riding, although this is obviously a subjective consideration.

A side note on non-original mounts

Each manufactured part has a set of dimensions. For example, the rails on the mount are x mm long, and they are y mm apart, and they are z mm thick. Also, a manufacturing process will produce parts with a certain variance around those dimensions. Sometimes the process is highly variable, and you often get, for example, rails that are too thick or too thin. The result of that could be that the eTrex is too hard to slide on, or that it slides on (and off!) too easily. We usually say the tolerances are poor.

With knock-off mounts, you do run the risk that a) they didn't exactly replicate the average dimensions of the rails and other important bits, and b) that their manufacturing tolerances are poor. I assume this is the core of @ChrisH's objection in comments. I agree with this in general, especially for cheap knock off components.

Of course, we don't know that Garmin's own manufacturing process for the eTrex mounts has that tight tolerances. All else equal, tighter tolerances should be more expensive to manufacture. Garmin may not be assuming that many people cycle with the eTrex. They may simply not have designed the mount to be very secure to begin with.

I have the same style of mount that I use with my Garmin Oregon 450. I have used the same mount for years of riding and I've rarely had an issue. I did have a couple issues in the beginning, but I realized this was due to my own problems and I would sometimes mount the GPS incorrectly, usually with only one of the side rails engaged, or not putting the GPS fully into the mount until it clicked. After I started ensuring that it was mounted correctly I haven't had any issues.

I have had issues with it turning around the bar if the zip ties weren't tight enough. In this case I actually chose to mount it to my stem which seems to be better because it can't rotate forward and back.

Perhaps they have started using a different kind of plastic causing it to wear out quicker, or there is some problem with the interface on your eTrex. The Oregon has a metal mount interface piece and from what I can see of pictures online the eTrex has a plastic interface piece. Perhaps there is some wear on your GPS unit itself that makes it more likely for the device to come out, or maybe just the fact that it is plastic causes it to not hold as well.

Depending on the size of your GPS, I have found these types of holders to work extremely well. Tested up to 250kmh on a naked motorcycle and while riding my MTB. Also took 2 years for the colour to start fading somewhat. The flexibility prevents a lot of the usual mechanical constraints of traditional holders.

If your GPS does not fit or if you cannot find an adequately sized one, you could 3D print it. It makes up for a fun little project and you can customise it to your needs.

An extremely simple solution has worked for me with exactly this Garmin mount and eTrex 30. Just stick a couple centimeter long piece of electric tape between the mount "rails". It presses firmly against the inserted navigator and completely dampens the annoying vibration.