The name "Dracula" is synonymous with vampires for most people. You'd be hard-pressed to find anyone, anywhere, who didn't immediately know who "Count Dracula" is. So it's hard to believe that when the novel first came out in 1897, no one had any such associations. "Dracula" was just a name to the early readers, just as "Count Dracula" had no particularly sinister associations to Jonathan Harker when, in the first chapter, we read in his journal that he's off to Transylvania from England to conduct a business transaction with a nobleman by that name.
Stoker actually got the name from a book about Transylvania he found in a library in Whitby (the town where Dracula first arrives in England) – a medieval Romanian named "Vlad III Dracula" was more commonly called "Vlad the Impaler" because of his habit of impaling in a pointy spear political rivals, religious dissidents, and pretty much anyone who crossed. Not a nice guy.
"Dracul" is Romanian for devil (or "dragon," originally), so it's an appropriate name for the vampire – but it was also the name of an order of knights in medieval Romania. Vlad the Impaler's father was a member of the Order of the Dracul, so his son took the name "Dracula" meaning "son of the Dracul."
However, Stoker didn't even plan on naming the novel after the vampire Count Dracula – originally, he planned on titling it The Undead, and only changed his mind at the very last minute before the book went to press. Titling the novel after its villain makes Dracula the center of the narrative – which, of course, he is.