Naming things is hard.
The tool was about uploading and downloading data specified with a URL. It would show the data (by default). The user would "see" the URL perhaps and "see" then spelled with the single letter 'c'. It was also a client-side program, a URL client. So 'c' for Client and URL: cURL.
Nothing more was needed so the name was selected and we never looked back again.
Later on, someone suggested that curl could actually be a clever "backronym" (where the first letter in the acronym refers back to the same word): "Curl URL Request Library"
While that is awesome, it was actually not the original thought. We sort of wish we were that clever though…
There are and were other projects using the name curl in various ways, but we were not aware of them by the time our curl came to be.
Soon after curl was first created another "curl" appeared that makes a programming language. That curl still exists.
Several libcurl bindings for various programming languages use the term "curl" or "CURL" in part or completely to describe their bindings, so sometimes you will find users talking about curl but targeting neither the command-line tool nor the library that is made by this project.
'to curl something' is sometimes used as a reference to use a non-browser tool to download a file or resource from a URL.