First of all, they would need an absurdly large directional antenna to even receive GPS satellites. GPS signals are already below the noise floor on Earth's surface, and we use mathematical trickery to extract them from the background. Secondly, even that would only be possible when Earth's over the horizon at the location of the probe. Thirdly, the whole span of GPS satellites as viewed from Mars would be a small fraction of arc second, i.e. basically all coming from the one place. Lastly, there's extra light lag that you have to compensate for, that depends on relative position and orientation of Earth and Mars.
No, GPS is not suited for use in space, especially not far away from Earth. However, I read once that some Earth orbit missions did in fact look / make use of the GPS signals, as in (low) Earth orbit, they're actually easier to receive than on the ground.
While Earth is in the Martian sky, and if the signal were strong enough, a viewer could average receiving signals from over half of the constellation rather than only half a dozen. Furthermore the time signals would originate from many of the satellites moving less orthogonally to the viewer and instead moving toward and away from the viewer.
Compared to measuring the sun relative to a changing horizon, it seems plausible that an extremely precise direction and altitude relative to Earth could give even more precise location on Mars.
Not contradicting you, but just to be clear: GPS satellites are not in geostationary orbits above the equator, or even in geosynchronous orbits. Rather, they are in medium-Earth orbits and their time-averaged density over the Earth is only a little lower over the poles as the equator. At some parts of the month, there are as many visible at the poles as at the equator. See Fig. 4 here:
https://www.researchgate.net/publication/228400966_Analysis_...
The GPS Wikipedia page also has a reasonable illustration of their orbits.
Not just timing, also space - Earth is a fraction of an arc second on Martian sky, pretty much a very small dot. All your signals would be coming from that small dot, i.e. practically on top of one another.
The difference in position between the satellites is at best 40,000 km (MEO orbits and satellites in opposite positions around Earth would give you the largest baseline), the difference from Mars to Earth is > 50M kilometers. So when viewed from Mars this would be like trying to triangulate your position in the United States based on signal sources spaced a very short distance apart somewhere in Moscow.
(sorry for the strained analogy)
Another problem with the approach is that of the two possible solutions that are the result from computing your location from GPS satellites one of the solutions is deep inside the Earth, which for both radio related reasons and reasons of practicality can be safely ignored. From a location in space very far away that trick no longer works so you will end up having to pick one of several answers.
Of course you could stick a GPS like transmitter into every Mars orbiter we launch from now on so that at some point there will be enough coverage locally to allow navigation, but that's a pretty expensive trick, besides that you'd also need a bunch of base stations in order to properly compute the orbits of the satellites to the required precision so that you can tell the satellites where they are.
So could we deploy, say, the first three satellites that use a modified version of GPS at first that allows you to get a very slow, inaccurate fix, but a fix nonetheless?
On Earth, similar systems were used for the TRANSIT satellites, and the Cospas-SARSat program. The latter is really cool, as it used weather satellites in low earth orbit to find a person in distress, using a cheap transmitter that's placed on a plane, boat, or very famously, Richard Branson's watch.
The way it works is that the device transmits a very stable frequency. (Perhaps modulated by the identity of the person in trouble part of the time.) As the satellite passes by, it relays the signal to the ground in a way that preserves the doppler shift. When you know the position of the satellite and the doppler shift, you can know the closest point and the distance from it.
After a few passes of your one satellite (or multiple satellites, if you have them), you can get a location that's good enough to start search and rescue.
Of course, time has passed since this was designed in the 70s and early 80s, and now GPS has fallen in price to the point where it's everywhere. So now, the devices send GPS coordinates with the identity info.
Mars is fifty million kilometers away. A GPS reliever on Mars looking for Earth satellites wouldn't be able to hear anything.
However, that does bring up an interesting idea. Maybe rovers could navigate using the position of the stars? It doesn't require you to launch 4+ GPS satellites into martian orbit.
Consider it like running. How far can you run in 1 second. That's the Moon. How far can you run in 21 minutes. That's Mars. It's 1260 times the distance.
