Does revealing Edge of Night from Hidden trigger Hold 'Em to reduce its cost for the turn?

Yes — playing Edge of Night from Hidden counts as playing a Gear to a battlefield, so Hold 'Em costs [2] less for the rest of that turn.

Two clarifications, since Hold 'Em doesn't actually "trigger": it's a static cost discount ("costs [2] less if you've played a unit or gear to a battlefield this turn") that's simply checked when you play Hold 'Em. And a hidden Gear is one of the few ways a Gear ever gets played to a battlefield — Rule 811.1.d.1 requires a hidden permanent to be played to the battlefield it was hidden at, overriding the normal "Gear goes to Base" restriction. Played from hand, Edge of Night would land at Base and wouldn't satisfy the condition.

The check is on the turn you reveal it, not the turn you hid it, and it looks at whether the card was actually played — if Edge of Night is countered before finalizing, it won't count (Rule 420.4.b). Note Edge of Night itself costs [0] when played from Hidden, so no discount ever applies to it.