I'm a caver. Things I look for in a lamp include:
• Robust. It has to stand up to hard use. Headlamps are usually the primary light source; taking two headlamps with you is weight and space prohibitive, therefor the main source of life has to stand up to hard use.
• Waterproof. Has to handle falling water and submersion. Unless you are are a diver--which takes us into a whole different spec world--the lamp generally won't be submerged for longer than a minute when say pushing a sump, or deeper than a meter.
• Flexible mounting/headband system. Has to mount easily to helmet or head. If there is something to keep the straps from moving on a helmet, that helps in tight passages. Should be easily adjustable, preferably while wearing gloves, while transitioning between helmet, head, and (insulating or rain) hat.
• Several lighting modes. If there is a need to bivy up for a while it's nice to have a low power setting that is good for days. Hi power should range between 500 and 1000+ lumens, with more being better as a rule of thumb as it's nice to be able to light an immense space when needed.
• Spot to wide focus ability. Wide beam should allow you to see your feet. Spot should allow you to see the top (or bottom) of a hundred ft. + pit.
• Batteries. Should be a common battery type, for the most part. Here it gets complicated. If caving in an area where resupply is easy, then more exotic battery types are okay, such as 123s or 18650s. The further off the beaten track you get the more common the battery type needs to be, i.e. AAs or AAAs. I'm not a huge fan of proprietary battery packs, with a couple exceptions. If they provide a significant performance improvement, and are inexpensive enough to purchase multiple backup battery packs, or there is some sort of adapter that allows you to use standard batteries if needed, and if recharging cables are common types rather than some sort of proprietary cable I'll think about owning one. Proprietary battery packs from a not particularly well established company are a concern as there is no guarantee they'll be available a year later.
• Weight and distribution. Less of an issue than back in the 4 D cell incandescent days, but still a consideration. Four 1860s installed in the lamp might make for a long burn or great performance, but could lead to a helmet wanting to rock forward and need readjustment, which gets old over a 36 hour trip. Placing the battery pack on the rear of the helmet helps balance things, but then the wire/connection has to be quite robust and designed in a manner that it doesn't hang on things. I'm not a huge fan of coiled battery pack wires as they are a lot more likely to catch on stuff/abrade.
I realize caving is a bit of a niche, but many of the design parameters mentioned would also serve other pursuits well, such as hardcore trekking, climbing, etc.