Although commenters Vince, Dave, Christopher have all been members since 1999, I'm afraid I can't declare a winner to the "oldest queue" contest, but I can say that the oldest Netflix membership you can have is from September 1999, which is when Netflix rolled out the Marquee plan, the first version of the subscription model. That was the first Netflix deal which involved having a queue. It wasn't until 2000 that they did away with the a la carte rental option and the Marquee plan went unlimited. I think that's when Netflix As We Know It began. They had just a few hundred thousand members, and only 7000 titles. 97% of us have joined since then. I can't believe it took me so many years to become a convert!
If you Google "Netflix Profile", you might run across a link to DW from MO, who has has been a Netflix member since Sept 1999, has rated 3305 movies, and written 230 reviews.
Even though Google indexes them, it looks like the Netflix Profile links aren't really public. I had to sign into my Netflix account to see FW from Garland, TX, aka spindaddydad.