Assuming I have applied Researcher Empowered in base, my opponent is holding BF1 with a unit that has deflect. I move in and attempt to stupefy the enemy unit.

Yes, you can play Stupefy, and your empowered Applied Researchers will offset the Deflect Power cost.

Here's how the costs work out:

1. Stupefy's base cost is [1] Energy.
2. Deflect adds a mandatory additional cost of Power equal to its Deflect value (e.g., +1 Power for Deflect 1).
3. Applied Researchers reduces the total cost of your spells by [1] Energy and [1] Power. Discounts are applied after all additional costs are summed (Rule 356.4), so it hits the Deflect Power cost too.
4. The "minimum of [1]" restriction applies only to the Energy portion — it doesn't stop the Power discount.

Final result with Deflect 1: You pay [1] Energy and 0 Power — the Deflect cost is fully offset. With Deflect 2, you'd still owe 1 Power after the discount.

Timing is fine too — Stupefy has [Reaction], so you can play it during the Showdown (Open State or even in response to other items on the Chain).