Blockading Charleston was a pretty outrageous action that earned him notoriety, and he had an epic end in combat with naval forces, which made his one of the most memorable stories in Captain Johnson's 'A General History of the Robberies and Murders of the most notorious Pyrates' (1724) which for centuries was the book on the early 18th century pirates.

Several other of the most famous pirates also had relatively short careers- Calico Jack Rackham 2 years, Charles Vane and Benjamin Hornigold 3 years, Black Sam Bellamy and Stede Bonnet 1 year.

This is likely because of the boom in Caribbean trade and surplus of trained, unemployed sailors following the end of the War of the Spanish Succession (since all of these were active between roughly 1715 and 1720) made it possible to become a rather notorious pirate in a very short space of time. The brevity of their fame was usually caused by either hanging, combat, or simply retirement once reasonable wealth had been acquired

I don't have much to add that hasn't already been said, but I'll see if I can consolidate a bit.

There are two answers to the question. Firstly, why Blackbeard was so infamous in his own day, and secondly, why Blackbeard is so infamous today.

In his own day, Blackbeard achieved a fair amount of notoriety because it's what he wanted. Probably more than any other pirate of the era he went out of his way to create a name for himself, and he was a first-rate self-publicist. Growing his beard was probably one of his greatest achievements: in 1717-18 beards were not at all common, if you look through a thousand pictures of men from the early 18th century you'll find maybe half a dozen beards (I speak from experience). It's not just that they were unfashionable, they were uncommon. Blackbeard is the only pirate of the 'golden age' whose beard is mentioned anywhere. By growing a beard he made himself unique, and simultaneously gave the public something with which they could readily identify him. Of all the pirates of the 'golden age', Blackbeard is really the only one with a nickname of that kind. 'Calico' Jack Rackham might also be considered, but the evidence that anybody actually used that name in his lifetime is slight, and it's still only an appendage to his given name: the only pirate whose name was replaced by a nickname was Blackbeard. That beard of his really was a publicist's dream.

His actions, particularly the blockading of Charlestown but also the burning of the Protestant Caesar also significantly added to his reputation as a man to be feared.

Finally, for reasons best known to himself, Captain Charles Johnson, in his General History of the Robberies and Murders of the Most Notorious Pirates (London, 1724), chose to devote a fair amount of page-space to Blackbeard, and so enhanced his already significant publicity. The first edition of the book carried three illustrations: Bartholomew Roberts, Anne Bonny and Mary Read, and Blackbeard. Johnson probably realised that such a larger than life character as Blackbeard would sell. Roberts was, at the time, probably the most notorious pirate, and Bonny and Read of course were women, so we can see that the illustrations were chosen carefully to have the widest possible appeal, and Blackbeard, with his beard, his nickname, and his reputation, certainly had appeal. Interestingly, Johnson may well have fabricated elements to enhance the Blackbeard story still further - the burning fuses under his hat can be found in no earlier source than the General History, for example.

However, despite all that, Blackbeard didn't remain a major feature of the popular culture of pirate into the present day. When Poe wanted a famous pirate to base a story around he chose Captain Kidd; Blackbeard isn't mentioned in Treasure Island (as far as I recall), even though lesser-known pirates like Edward England are, and despite the fact that Stevenson's character Israel Hands borrowed his name from one of Blackbeard's real-life crew.

Blackbeard as a character seems to have enjoyed a resurgence in popularity in the second half of the twentieth century. In 1952 Robert Newton starred in Blackbeard the Pirate, in 1956 Blackbeard was a demonised character in the TV series The Buccaneers, and from then on he was a staple of pirate drama and fiction. His return to the forefront probably owes much to his original notoriety, coupled with Johnson's emphasis of him, but for a long time he was a footnote to others, particularly William Kidd.

Oddly, the popularity of pirates in film and fiction seems entirely unrelated to their real-life success. The most successful pirates of the 'golden age' were men like Richard Taylor, Henry Every, Thomas Tew, and Oliver la Buse, who rarely if ever get a mention outside history books.

In addition to the factors already mentioned by others, it should be noted that Blackbeard himself was a very impressive man who did his very best to utterly terrify his enemies, especially in combat. He would go into battle with cannon fuses smoking in his beard and hat, appearing like the Devil himself. As such, it's easy to see why anyone who met him, even if there weren't that many of them, would be very impressed and tell tales about him, increasing him infamy.

I'm going to go ahead and page our own /u/ETFox here if he's around, since this seems to be right in his balliwick. He's written on Jacobite pirates before, including Teach, and would be one of the best placed to understand the various factors at play here. He's written a short paper that's pretty accessible on this, but unfortunately I'm on limited hotel wi-fi and they don't want me to be downloading PDFs. But you can easily find it by searching "Jacobite pirates" and E.T. Fox in google. It doesn't directly answer your question, but should go a ways to understanding the context.