Linguist Benjamin Zimmer reveals the whole shocking story:
[T]he brand name was coined by the inventor of the device, NASA scientist Jack Cover. Cover reportedly formed the name from the initial letters of Thomas A. Swift’s Electric Rifle, after the 1911 juvenile adventure novel Tom Swift and His Electric Rifle. Since the protagonist Tom Swift never actually revealed his middle initial, we can guess that Taser is something of a backronym, i.e., a word that is “retrofitted” with an acronymic expansion after the fact. Taser appears to be modeled on an earlier acronym, laser (”light amplification by the stimulated emission of radiation”), which in turn was modeled on maser (”microwave amplification by stimulated emission of radiation”). Another possible inspiration for Taser is phaser, the name of a fictional weapon familiar to Star Trek fans. According to Jeff Prucher’s Brave New Words: The Oxford Dictionary of Science Fiction, Star Trek creator Gene Roddenberry originally wanted to call the weapon a laser but then opted for phaser instead.
The Taser went on the market in the early 1970s, but it took about 15 years for a verb--to tase--to emerge as a back-formation. (Previously, "to Taser" had been preferred.) According to Zimmer, it was the Rodney King incident, in 1991, that brought tase into wider circulation.
It's interesting to think about why some trademarks, such as Taser, are back-formed into lower-case verbs, while others, like Google, retain their full capitalized spelling: no one says I googed her.