The word for the biggest machine humans have ever built started as an adjective in a technical memo. Nobody sat down to name it.
The pieces are ordinary. Etymonline breaks the word into inter-, "between," and network. Network itself is far older, from the 1550s, when it meant a net-like arrangement of threads or wires. By 1839 it could mean any complex, interlocking system.
In the early 1970s, engineers were trying to connect separate computer networks so that a message could cross from one to another. Etymonline says they first wrote about internetwork or inter-network from 1972, describing hypothetical networks joined together. The longer word was then cut down to internet.
Vint Cerf, one of the designers of the protocols, has said he thinks the first use of "internet" was RFC 675, in December 1974. That document, by Cerf, Yogen Dalal and Carl Sunshine, was titled Specification of Internet Transmission Control Program. In that document the word works as a modifier, as in "internet packet," not as the name of a place.
The usage was loose for years. Joseph Reagle, who surveyed the early protocol papers, found "international," "internet" and "internetwork" used "rather interchangeably throughout the 1970s." A rival term also existed. Louis Pouzin coined catenet in a 1973 note, and it appeared in technical writing into the early 1980s.
The short form won. According to Wikipedia, catenet was gradually displaced by internet as the Internet Protocol spread from the mid 1980s. Etymonline dates the sense of the linked computer networks of the U.S. Defense Department to 1984.
That shift turned a common noun into a proper one. An internet was any network of networks. The Internet was one particular network, with a capital letter. The capital has been fading. Etymonline notes that the Associated Press style guide began lowercasing the word in 2016.
Other early "internets" existed too. A November 1977 memo from the computer maker Datapoint planned the launch of a product called INTERNET, software linking terminals and processors by coaxial cable. It shows how naturally the word suggested itself to anyone joining machines together.