I'm slightly puzzled by the photograph and subsequent comment.
First off, looking at the photograph I'd say the source of your problems 'aint with the resistor. Have you actually had a meter across it and confirmed that it has failed?
The reason for my statement comes from two observations.
1) The conventional, through-hole, resistor is sitting on side 'A' of the printed circuit board, with no signs of burning beneath it. The opposite side of the circuit board (side 'B') has extensive burn marks.
Question: If the resistor got hot enough to burn the board to a crisp on side 'B', why hasn't it marked side 'A' sitting right next to the (very hot) component??
2) When a resistor gets hot enough to burn the circuit board it is getting VERY hot. 'FR4' pcb material starts to burn at around 130 deg. C. I don't know what material speleotechnics use, but it gives you an idea. When a resistor has been operating at these temperatures for extended periods of time (ie. long enough to cause extensive burning of the pcb) it usually becomes a bit discoloured itself. On dark resistors this might not be too noticeable, but on beige or pale blue ones it's often fairly clear. The photograph shows a perfect looking pale blue resistor.
This still leaves the question as to what caused the problem. The comment:
the varnish type coating around the component has come away leaving the metal underneath.
The photograph clearly shows the component and clearly shows no damage on the component side. The speleotecnics 7 & 14 LED headsets I've seen don't use surface mount resistors (in fact the first ones I saw didn't have any form of current limiting device!), so the other side of the board just features the tracks.
For the board material to burn there must have been something hot in close contact with it, this could either be in the back of the headset, OR it could be the pcb track itself. I also note that the photograph shows a dollop of hot-melt glue. If this stuff is in the area then it will stink to high heaven if it gets burnt (- trust me, I nearly evacuated our whole lab when a diode gave up the will to live under a coating of hot-melt!). Not to mention the pcb material itself, this smells awful as it burns! In comparison an overly hot resistor isn't too obnoxious.
From the photograph and comments I'd say the most likely cause is a short circuit somewhere that has drawn exessive amounts of current through the pcb track, causing it to heat up and burn the pcb. Unlike Oldhams, the speleotechnics system doesn't feature any form of over current protection to prevent damage.
Where the short circuit came from is the next puzzle. As already mentioned, I think you can discount any water. This may lead to electrolytic corrosion, but at 3.6V it's not going to cause huge currents to flow. From the comments the more likely cause could be the nicks in the cable. If these have been there for some time and have gone unnoticed then it may explain the problem. Having installed one of those reflectors on someone elses behalf I already know it's a tight squeeze and you can't guarantee where the wires are going to lie.
On their own the nicks are unlikely to cause a problem, BUT the switch mechanism inside the headset consists of bare strips of moving metal. I think the most likely cause of your problem is that one of the nicks in the cable has come into contact with possibly the switch (or other terminal) and has created a short circuit. I seem to remember the tracks on the LED units are weedy thin things so will quickly heat up with excessive current flow. In comparison, the nicked cable and the terminals are both substantially thicker and able to handle the current without signs of damage. As the track heated up it burnt the PCB and quite likely fused it, hence why half the LEDs have stopped working. I haven't checked, but I think there's only one resistor for all the LEDs, so if the resistor had blown ALL the LEDs would stop working, whereas the original comment said only 7 out of the total 14 had "gone west".
OK, it's only a theory, to prove it one would need to see a photo of the other side of the board and to see exactly which wires are nicked and how large the nick is!
Cue next suggestion....
Cheerio,
Mine X.
PS: Just looking at the photograph again, by the look of it, the resistor sits at either the 3 O'Clock or 9 O'Clock position when it's in the headset - ie. 50% chance it's directly over the switch mechanism...