Can you bounce your own permanent in response to an opponent's counterspell to save it?

This is a common misconception. When you cast a spell, it goes on the stack as a spell object. A counterspell targets and interacts with that spell object on the stack — it does not target any permanent on the battlefield (CR 608.2b). Bouncing a permanent returns it from the battlefield to your hand, which is an entirely different zone and a different object.

Because the spell on the stack and any future permanent it would create are treated as separate objects (CR 400.7), removing a permanent from the battlefield has no effect on a spell that is already on the stack waiting to resolve. The counterspell will still resolve and counter your spell normally.

The only ways to save a spell from a counterspell are: (1) give it hexproof or make it uncounterable (e.g., with Veil of Summer), (2) counter the counterspell itself with another counterspell, or (3) resolve the spell before the opponent can act — none of which involve bouncing permanents (CR 702.11, 509.2).

Concrete example: You cast Llanowar Elves (it's on the stack). Your opponent casts Cancel targeting it. You activate a bounce ability returning your Elvish Mystic (already on the battlefield) to your hand — this does absolutely nothing to protect Llanowar Elves on the stack. Cancel resolves and counters it.

Note: Bouncing a permanent already on the battlefield can be useful to save it from removal that targets a permanent (like Doom Blade), but that is an entirely different situation from protecting a spell on the stack from a counterspell.