I've worked with 'em all of my life, in many different technologies,...and can say that this particular app (DME) would be the simplest environment. There are several specs that differentiate one from another, with bandwidth being the most significant. One would do just fine with a 10MHz BW in this application (ie: a 500 MHz BW would be overkill (and higher in cost) in this app,..although that's what I have and would easily be up to any DME tasks). Sensitivity would be another spec that you would pay for (no need for extreme sensitivity),...# of channels, yet another spec (single channel would do fine I would imagine).
Another: digital vs. analog,..analog would be fine... Any of these specs would differentiate one scope from another,...the more the merrier!!!
Now if one were to get into looking at the processor signaling, you'd want to be aware of clock frequencies (BW),......yet still this is quite primitive circuitry, although I'm sure the people that repair these (Loren, etc.) could chime in with additional comments as to the digital side of things. Also, one could easily see any noise being generated, to and fro. These signals really are quite simple insofar as "goes into, goes outa".
NOW, when one delves into the circuits (processing of these signals internally), this WILL require a degree of electronic knowledge in order to troubleshoot/repair,..regardless of whatever test equipment you may have in your hands.
Handheld units are quite nice in that one wouldn't have to drag a bench device out to the car but would be a preference,..price could be outside of one's budget.....what can you afford?
Used scopes can be had quite cheaply and would be a good bet for this app (ins/outs). One would want to know the device is calibrated properly but extreme precision wouldn't necc. be a factor here with the DME,,,,,,accuracy: yes.
Maybe we can start to lay out some sources for this piece (used or new)..
http://oscilloscope-tutorials.com/
http://science.howstuffworks.com/question8.htm
http://www.williamson-labs.com/scope-main.htm
http://www.apogeekits.com/oscilloscope_hps40.htm
http://www.electronickits.com/gold/hps10.htm
my 2 cents....
Best,