Quick Claw doesn't have +1 Priority it just enables a Pokémon to move first in it's priority bracket. It also doesn't give +8 Priority that is only the triggering message saying "< Pokémon's > Quick Claw Activated" or whatever.
>A hold item that gives the holder a 20% chance of going first in its priority bracket and ignores the effects of Trick Room and Stall.
>When a Pokémon uses a move, some held items and Abilities cause it to move either first or last in its priority bracket, ignoring even Trick Room. The items Full Incense and Lagging Tail and the Ability Stall cause the affected Pokémon to go last in its priority bracket, while the items Quick Claw and Custap Berry cause the user to go first in its priority bracket.
So for example if I had a slow Pokémon... I don't know... a Level 100 Gigalith with a Quick Claw as it's held item, and my opponent had a Level 100 Mega Charizard Y (who was much faster than Gigalith). If Gigalith uses Stone Edge and Mega Charizard Y uses a move of equal priority for instance Solar Beam, Gigalith would go first if Quick Claw were to activate.
>Quick Claw is the most common one you will see. It has a 3/16 chance of boosting your Pokémon's priority and attacking first. However, this is not utilised in skipping priority stages. So if you have a Quick Claw attached and it activates while you use Flamethrower, but your opponent uses Quick Attack, you will still attack second.
Source: Knowledge and Experience.
Additional Sources: Bulbapedia - Quick Claw | Bulbapedia - Priority | Serebii - Priority