Or maybe “Arrrrrrchie”
Jimbo, in honor of the SDMB’s own Dread Pirate.
Henry? (After Sir Henry Morgan)
Bartholomew? (After Bartholomew Roberts AKA “Black Bart”)
Calico Jack?
Swabby
Ahab
Plunder
Loot
Booty (this might get you some strange looks at the park)
Spot (as in “X marks the…”) ok, that was really weak
**
Jonah** (though it might be unlucky 
)
“Swag.” Combination of “sweet” and “wag.”
Cap’t. Liftleg
Cap’n
Westley
Sinbad (Ok, he was a sailor, not a pirate, but it fits)
Swashbuckler
If you like those long drawn out names you could call him
Avast Ye Swabs, Prepare to Be Boarded
and then just call him Vasty
I was going to suggest that. It would sound rather strange calling the dog though, “Heeerrrree Arrrrrrrrrrrrrrrrrrf!”
I vote for Matey. It fits right in to pirate-y conversations.
“Arrrr, Matey, so it’s liver you’ll be havin’.”
The pic looks like th dog is red or at least reddish. if so, Redbeard fits perfectly. If he develops a tase for the local wildlife, he may even come home extra red-bearded a time or two.
Darrgh, matey!
Poop Deck.
At least until he’s housebroken…
Bootstrap Bill
I have always had a soft spot in my heart for the word Scalliwag.
But not such a soft spot that I am actually sure of how to spell it.
Matey is also both a good dog name and pirate word. So add my vote to Matey.
Quohog
But of the suggestions on here so far, I like “Swag” the best.
And he’s cute!
Before I opened the thread, when all I saw was the title, I was thinking Arrrr or Arrrf.
Seeing his picture, it’s so obvious…
Redrum!
Captain Jack McGilliwhiskers, yarrrrr!
Dear god, I don’t think we’ll be able to avoid referring to swabbing his poop deck.
The answer is:
Bounty