To keep this simple....lets call the "relay" the "solenoid". I'm old school, and printed words tend to confuse me. As there are 3 relays also under the "faux oil tank" electric cover. One of which also deals with starting.
Don't let this confuse you. The solenoid, is under this cover too.
Read all these instructions while next to the bike before you jump in, at least once. Make sure you know what I'm writing about here. It'd be more likely a bit easier if you have a copy of the parts catalog to look at. (its out there on CD, lemme know if you'd like a copy, I'll burn one, you pay for postage)
Unless you've done some customizing on that cover (I don't recommend) , you will be attempting a bit harder job than a normal simple fixes.
This does sound like a possible solenoid issue, but I strongly suggest checking battery connections at both ends. As suggested above, it could be the ground. Notoriously, the factory placed that bolt with loctite, and that product isn't very conductive. Either way, you'll have to pull that cover off.
I've had many a tussle with the battery cables on the battery. I cannot stress enough, that they maybe corroded, or not TIGHT enough. It can be surprising with just a touch more tweak and the problems go away. If your X still has the factory cables on it, I'd seriously look at getting the updated ones. The OEM cables are just too stiff and too short.
To access that cover, you'll need to pull the rear head exhaust pipe/header and heat shields. The cross over pipe can be challenging....just be patient, and fool around with it, it'll come off. It can be done on the ground, but is much easier on a lift (I've got a cranky back).
Then you'll have access to the cover. I think there are 4 allen heads/button heads holding it on,.... one is down low and easy to miss on the left side. Make sure you pull the dip stick before removing.
Once that cover is off, you'll be able to see everything. Mounted on the left side is the solenoid... it'll have a red rubber cover over it. Hopefully its just a loose nut. Good time here to check tightness on those battery cables.
Inside this area of the electrical box, on the bottom, just right of center, in the case is the grounding bolt. You'll see 2 way too small ground wires attached to it. Since you are there, if it hasn't been done already, take both of those grounding wires off and replace with something bigger, with less resistance (I strongly recommend this). Also clean that bolts threads while you are at it, and the threads into the case.
After you've checked all that, give it a whirl, and try to crank the motor. You can do this either by turning on ignition and hitting the button or jumping across the low amp poles on the solenoid, with the igntion switch off. Is X in neutral... umm?
When doing this test, one of those 3 relays on the back of box, should click if ignition switch is on.
If the solenoid appears to test bad.... Give Mike Seastrome at EHparts.net a call. Last I knew there were a shortage of OEM left. If he or the late Jamie Jones haven't come up with a good replacement, I'd strongly suggest going to your local lawn mower shop and show/tell them what you have got. This has been done successfully in the past.
If this is all well and good, you'll be on the hunt for a direct short somewhere in the wiring an up to the starter button, and/or the ignition switch.
This may all sounds foreboding and scary here in print. It really isn't as bad as this appears. Just takes time and patience..... reserve sometime to do this, with no urgency.
Every step listed above, take a shot cranking it over. In jumping down this road, its very possible it was something simple you did along the way.
good luck, and let us know how it went. If any of this is confusing, come on back. Greg, Mike, and I are more than willing to help any X owners out there cornered.
Our common goal is to keep these beauties on the road.
Bruce
flatlanderhelpingXworld
Ive edited this post/answer about 6 times... I think I've got it