In Witness theology it has to do with the notion of answering the devil's challenge seen in the beginning of the book of Job. Everyone can stand up in "the universal court of divine sovereignty" (organizational phrase), that of answering the accusation which Satan made that all intelligent beings only serve God for the personal benefit gained. Each one then has the opportunity to be one of Jehovah's Witnesses, not a Jehovah's Witness.
As a parallel you might consider a real court case: you have a friend named Sally who has been unjustly accused of something. You appear on behalf of your friend as a character witness. Would you be a Sally's witness or one of Sally's witnesses? Like it or not, that's the thinking behind the term "one of Jehovah's Witnesses."
It's also why they say that Abel, Abraham, David, and Jesus were individually one of Jehovah's Witnesses, not a Jehovah's Witness.
Of course, there is the secondary reason mentioned above, that the Witnesses want so greatly to separate themselves from other professed Christian religions. The primary reason, however, is theological.
I've got to include this funny story. I was once on a shepherding call with the CO, a call on an inactive brother. The CO asked him, "Do you consider yourself a Jehovah's Witness?" The inactive brother said, "I'm surprised to hear you use such a term that does not agree with the slave's publications. No one can be 'a Jehovah's Witness.' Isn't that the term used by those who don't understand the issue raised by the devil?" The CO was really embarrassed and about crapped his pants. I loved the turning of the tables on that one! I had to restrain myself from laughing.