What does the headlamp do?

The headlight itself lets you see better in dungeon area, but it is not mandatory really, you can even simply increase your screen brightness for a similar effect.

But the Spelunker upgrade also grants your miner the passive ability of locating rooms with treasures, this ability doesn't cost mp and has nothing to do with headlight.
So even though headlight isn't really useful the Spelunker upgrade is probably worth buying.

If you happen to have a Spelunker with glaucoma (Or whichever thing causes your vision to be dark wherever you go), the headlamp will remove this no matter where you go, essentially removing their detrimental trait if you remember to press A.

The headlamp does actually have one use!

You can use it in the "Dungeon Area" or the "Fourth" part of the game; to light up the area. It's dark down in the cellars afterall and a little light can help slay a few enemies along the way! ;D