In a group of people the white rabbit was the person in charge of keeping track of time. In fact, if you watch the movie Resident Evil with the cast commentary on, there's an interesting story about how each character portrayed a different character from Alice in Wonderland. The character portraying the white rabbit was a member of a swat team-type force who was skilled in computers and kept track of the team's evacuation time. When I was in Korea we had a curfew by which time we needed to be back onto the military installation. As people drinking heavily on a weekend, it was easy to lose track of time, so we would elect a white rabbit to ensure we left the club on time. I was usually the one chosen since I was extremely punctual anyway.
During the same period I was a assigned to an air-defense unit. Our mission was to defend a military airbase from North Korean missiles. It was my job to relay messages between the radar and the command center. I had to keep track of four separate time zones, and numerous air space corridors that were friendly to fly in only during certain times of the day. The job required me to wear two wrist watches and carry a stop watch. Again I held the title of white rabbit, but it was still just a reference to my position and not me personally.
At the same time I was hosting secret gatherings in my room (known to us as the Zen Room) on the weekends where me and several of my friends would take dangerous amounts of dextromethorphan and lie around the room in a drug induced state listening to weird music and having even weirder conversations. We would seal off the windows and sit in pitch blackness since the light hurt our dilated eyes and the darkness brought on stronger hallucinations. One of the many house rules was that only one person (usually me) was allowed to wear a watch since everyone was stuck in 'acid-time' and ten minutes felt like hours. When someone wanted to know what time it was, they would call out "WHITE RABBIT", then the time keeper (again, usually me) would tell them the current time. Then we would all talk about how weird it was and how it felt like hours had gone by. This went on for a while.
Most of my friends at that time didn't wear a watch to work and didn't need to because they knew that I had several. I was asked many times a day to give someone the time. Some of my friends from the Zen Room would call out WHITE RABBIT while at work if they wanted the time. This caught on really quickly even to people who knew nothing about the Zen Room or it's connection to me. They just knew that when they said "white rabbit" I would give them the current time. Many assumed it was my nick-name. Later when everyone in my unit was assigned a unique call-sign for communication over unsecured radio, it was no surprise that my call sign was "The White Rabbit." I have been stuck with it ever since.