Taser is an acronym for a book that most users have never heard of.
Jack Cover was a researcher who had worked at NASA. He developed the device between 1969 and 1974. Gizmodo says he patented and named it in 1974, though Etymonline records the word in 1972, so the exact date of the naming is unclear. His inspiration was a children's adventure series. The tenth book was Tom Swift and His Electric Rifle, published in 1911. Its hero fires electricity instead of bullets.
Cover wanted to call the device Tom Swift's Electric Rifle. The initials spell TSER, which does not make a catchy word. So he is said to have added a middle initial, "A," turning the hero into Thomas A. Swift. The initials then read TASER. Writer Mark Forsyth called it "a gratuitous initial."
The extra letter has no meaning of its own. Cover simply wanted a better word.
The OED, quoted on the blog Languagehat, calls it an acronym of the initial letters of Tom Swift's electric rifle, formed "after laser." Etymonline agrees and records Taser in 1972. Steve Tuttle of the later company confirmed in writing that "the reference to Tom Swift is correct."
The first model sold used gunpowder, and in 1976 the Bureau of Alcohol, Tobacco and Firearms classified it as a firearm. Rick Smith and his brother Thomas founded a Taser company in 1993 and worked with Cover on a version that the ATF later said was not a firearm.
The name has since escaped the brand. English now has the verbs taze and tase, with tased and tasing, and the company works to keep Taser capitalized.
One commenter on Languagehat suggested that Star Trek's phaser may also have influenced the sound of the name. No source supports that.