Can you counter your own spell to prevent an opponent's copy effect?

This depends on timing. When an opponent uses an effect that copies your spell (such as Twincast or Strionic Resonator targeting a triggered ability), the copy is placed on the stack immediately as part of resolving that effect. By the time you could counter your own spell, the copy already exists on the stack independently.

Countering your original spell will not remove the copy. Under CR 706.10, a copy of a spell on the stack is itself a spell and is only affected by effects that target it specifically. Countering the original has no bearing on the copy's existence or resolution.

If you want to prevent the opponent from ever getting the copy, you would need to counter the copy effect itself (e.g., counter their Twincast) before it resolves, not your own spell afterward. This is the correct line of play if your goal is to stop the copy entirely.

You can choose to counter your own spell for other reasons — for example, to prevent its effects from resolving if you no longer want them — but you should understand the copy will still resolve unless separately dealt with (CR 608.2b, 706.10).

Example: You cast Lightning Bolt. Your opponent responds with Twincast, copying it. You then cast Counterspell on your own Lightning Bolt. The copy of Lightning Bolt from Twincast is still on the stack and will still resolve, dealing 3 damage to whatever it targeted — only your original Bolt is countered.