Galileo did not name the telescope. For the first few years of its life, the instrument that changed astronomy had no settled name at all.
Spyglasses appeared in the Netherlands first. In December 1608 the Dutch authorities turned down Hans Lipperhey's request for an exclusive patent on one, even though it had worked well in tests. Galileo built his own versions and turned them on the sky. His Sidereus Nuncius, published in Venice in 1610, described what he saw.
People called the device many things. A review of Galileo scholarship in the Los Angeles Review of Books lists "eyeglass" and "long-sighted cannon." In Italian it was Galileo's occhiale. In Latin he wrote perspicillum, and historian Edward Rosen notes he put it on his title page. The same review notes that the instrument "would not be called a 'telescope' until 1611."
That year the name was invented in Rome, among the Accademia dei Lincei. This early scientific society was founded in 1603 by Federico Cesi and three friends, who named it after the lynx. On April 14, 1611, Galileo showed his instrument to a gathering of scholars there. The Vatican Observatory notes that the academy elected him a member eleven days later.
Who said the word first is less certain. Wiktionary and Wikipedia credit Giovanni Demisiani, a Greek mathematician and theologian from Zakynthos and a member of the academy. Wikipedia's only source for that claim is a 2006 newsletter from an amateur telescope club. The Vatican Observatory says only that a guest who was an expert in Greek invented the word.
Etymonline instead says Prince Cesi, the academy's head, is credited with coining it. Rosen quotes John Faber, who listed the dinner guests, writing that Cesi "thought up the name telescope." Both accounts agree on the year and the circle.
The coinage was built from Greek: tele, "far," and skopein, "to look." The result echoes the Greek teleskopos, "far-seeing." Etymonline records Italian telescopio in Galileo's writing in 1611 and Latin telescopium in Kepler's in 1613.
English took the Latin form from 1619, and telescope itself from the 1640s. Before that, Etymonline notes, an English observer might use a "trunk," short for "perspective trunk."
The word then grew a verb. Because some telescopes had tubes that slid inside each other, telescope came to mean "force together one inside the other" by 1867. The -scope ending also became a pattern for naming instruments, as in kaleidoscope.