The technical generic term for neon tubes is gas discharge tubes, or more precisely: cold cathode fluorescent lamps (short: CCFL). No wonder, then, that the colloquial name "neon" became established after the inert gas originally used in them. Incidentally, the name comes from the Greek word for new: νέος (néos).
In fact, most modern luminous signs no longer contain (only) neon, but its neighbouring gas argon (or a mixture of both). The reason for this is that argon is more reactive than neon, and thus less energy is necessary to make it glow.
Theoretically, all naturally occurring noble gases react to electricity with colored discharge: helium glows orange-yellow, krypton white or blue, xenon blue, and the radioactive radon red. The natural colour of neon is a bright red and that of argon is a pale purple/pink or light blue – hence our neon colours Classic Red and Classic Blue. Nowadays, only the latter two gases are primarily relevant for the production of neon signs.
Incidentally, neon gas is also used in neon glow lamps (sometimes just known as “neon lamps”), which are another type of gas discharge tube.