When I equip B.F. Sword to my Unsung Hero, he becomes and I exhaust Fiora. My opponent reacts and plays Stupify to the Unsung Hero. Do I still channel one rune exhausted?

Yes, you still channel 1 rune exhausted.

Fiora's triggered ability fired the moment Unsung Hero became Mighty (2 + 3 from B.F. Sword = 5). That trigger is now on the Chain. When it resolves, the effect is simply "you may exhaust me to channel 1 rune exhausted" — there is no re-check of whether the unit is still Mighty at resolution. Stupefy reducing Unsung Hero's Might to 4 doesn't stop the channel.

This is consistent with how Riftbound treats trigger conditions vs. effects: the condition is evaluated when the trigger would go on the Chain (Rule 382). The effect executes what's written — and Fiora's effect doesn't say "if that unit is still Mighty." (Compare Unsung Hero's Deathknell, which does re-check — "If I was Mighty, draw 2" — and would fail here per FAQ #4157.)