Contrary to popular fiction, pirates usually did not have "pirate names", they were known by their birth name. To survive harsh sea life, even pirates needed strict discipline on the ship. In military like atmosphere, it was not really advisable to call Captain, First Mate or Quartermaster by anything but "Sir," especially not by their "nickname."
In some rare instances, pirates like Blackbeard took their nickname and used it every day. Others like Samuel Bellamy had a lot of nicknames like Black Sam Bellamy, Robin Hood of the Sea, Prince of Pirates, Black Bellamy, but never actually called themselves by those names.
Some theories would suggest that pirates used nicknames to prevent government officials from identifying them, and perhaps from prosecuting their families. However, a more realistic explanation is that pirates even in the 17th century were somewhat romanticized and that the writers of that time like Captain Charles Johnson, who wrote "A General History of the Robberies and Murders of the most notorious Pyrates" used nicknames to exaggerate the menace that pirates were.
Fictional pirates, on the other hand, have a much more interesting origin of names. Captain James Hook from Peter Pan and One-Eyed Willy from Goonies are classic examples of fictional pirate lore and names that were popularized by Treasure Island novel and famous Long John Silver. Others like the Dread Pirate Roberts from the movie (and novel) The Princess Bride are variations of mostly English names and surnames with "fearful" adjective.
If you want to find you perfect pirate name, look no further than most common names in England during the 16th century.
16th century Men's popular names:
John, Nicholas, Leonard, Samuel, Arthur, Thomas, Ralph, Martin, Allen, David, William, Christopher, Simon, Charles, Fulke, Robert, Anthony, Peter, Alexander, Luke, Richard, Matthew, Philip, Gregory, Mathias, Henry, Edmund, Stephen, Nathaniel, Tobias, George, Walter, Lawrence, Abraham, Isaac, Edward, Hugh, Roger, Barnaby, Jerome, James, Andrew, Daniel, Geoffrey, Joseph, Francis, Humphrey, Michael, Reynold, Rowland.
16th century Women's popular names:
Elizabeth, Isabel, Christian, Barbara, Julian, Joan, Dorothy, Edith, Rachel, Philippa, Margaret, Margery, Emma, Charity, Audrey, Agnes, Susanna, Lucy, Mabel, Helen, Alice, Ellen, Martha, Millicent, Janet, Anne, Sarah, Marion, Rose, Sybil, Mary, Clemence, Cecily, Thomasin, Ursula, Jane, Frances, Frideswide, Fortune, Avis, Catherine, Joyce, Grace, Gillian, Beatrice, Elinor, Bridget, Amy, Judith, Blanche
With a combination of classic English surname like:
Smith, Jones, Taylor, Brown, Williams, Wilson, Johnson, Davies, Robinson, Wright, Thompson, Evans, Walker, White, Roberts, Green, Hall, Wood, Jackson, Clarke
Or perhaps occupational surname like:
Archer, Bailey, Baker, Brewer, Butcher, Carter, Chandler, Clark, Collier, Cooper, Cook, Carpenter, Dempster, Dyer, Farmer, Faulkner, Fisher, Fletcher, Fowler, Fuller, Gardener, Glover, Hayward, Hawkins, Head, Hunt or Hunter, Judge, Knight, Miller, Mason, Page, Palmer, Parker, Porter, Potter, Sawyer, Slater, Smith, Stringer, Taylor, Thatcher, Turner, Walker, Weaver, Woodman and Wright
You can get believable pirate names like Captain Nicholas Archer. If you more interested in fictional like pirate names, try various combinations of colors animals, adjectives, and everyday objects.