Wow, excellent topic!
In terms of character, I'd say a good referee has confidence - a clearly confident referee is likely to gain the respect of the players, and have the poise and courage to make the tough and tricky calls when required.
There are a lot of important qualities, as well as technical issues that developing referees should be aware of, and I hope that plenty of them will be discussed in this thread.
For the moment, however, I want to look at it from a slightly different angle.
A referee is expected to ensure players play by the rules of the game... to perhaps be a little lenient to novices but to require high standards from all other players. He or she is expected to judge the intentions behind a player's actions, ruling (and perhaps penalising) accordingly.
A referee is expected to be able to earn the respect of the players and be someone they can turn to for advice (where appropriate e.g. rules clarification). He or she is, ultimately, trying to create a sacred space for the duration of the game... a space within which the players have the opportunity to compete at their best (whether or not they meet this challenge)... and to do this the referee must ensure the space is one in which the players honour the spirit of RPS in their play.
Hopefully no-one would take issue with that description. But let us think... outside of competitive matches, who is it who fulfils the same role?
Encouraging novices... challenging players to excel... demanding respect for the game and punishing illegal play... holding the trust and respect of the players... clarifying rules or other confusing points... embodying the spirit of RPS in their actions... surely this could be a point-by-point job description for an RPS trainer/mentor/tutor?
One of the differences between good referees and great referees is that great referees implicitly recognise this and act in this way. (Just today, for example, Mr. Fox posted an encouraging message to a novice bullboarder.)
To summarise (about time, I hear you say), when players are training, their trainer is their master. When players are no longer training, the game itself is their master. And in competitions, the referee is the living embodiment of the game.