Time And Relative Dimension In Space TARDIS
The TARDIS was named by Susan, the Doctor's granddaughter, who had a liking for Earth, especially London, which is why the name is formed from the initials of a phrase in English. See "An Unearthly Child", originally aired on 23 November 1963.
Strictly, TARDIS is the name of the Doctor's ship only but it has come to be applied to other ships of the same kind. This is similar to the way the name Dreadnought (i.e., HMS Dreadnought, launched in 1906) has been applied to other all-big-gun battleships. Thus SMS Bayern can be described as "a German dreadnought of WW I".
The correct generic name for the time-space ships is TT Capsule (probably abbreviated from Transdimensional Time Capsule). In "Blink", when Sally and Larry finally get into the TARDIS, they trigger a holographic recording of the Doctor saying, "This time capsule has detected an authorised control disk..." In "The Sound of Drums" and "The Big Bang", the Doctor complains about "time travel without a capsule".
Presumably, TARDIS was adopted as a generic term by many species because the TARDIS was the first ship of its kind they ever encountered and was later taken up by the Doctor and other Time Lords because (a) many species used it and (b) it's more elegant than "TT Capsule". As time travel is involved, the term TARDIS could spread backwards in time, as well as forwards.