In Death Note Rem liked Misa so much, or maybe he was in love with her. He wrote L's name in the Death Note and killed L because he was protecting Misa and he knew he will die soon after doing so.
But I just wonder: why didn't he instead tell L's real name to Light Yagami or Misa, so they could write L's name in their Death Note and kill him themselves?
One of the Shinigami rules (shown during scene breaks and not the actual story) says:
The Shinigami must not tell humans the names or life spans of individuals he sees. This is to avoid confusion in the human world.
Another rule states:
If a Shinigami decides to use a Death Note to end the life of the killer of an individual it favors, that individual's life will be extended but the Shinigami will die.
While there is no specific consequence stated for breaking the first rule, one might also assume that it would result in similar punishment. In the end Rem probably decided to write the name herself due to the urgency of the situation. (And for dramatic purpose.)
Although there is a rule for a Shinigami not to tell a human name, it is not clear what would have happened if she did break the rule. Is it just a recommendation or is it punished by death? It seems difficult to believe that the punishment would be death as it does happen to say a name they have seen when they chat, either because the human already knows the name or because it sounds like a common word.
Though she didn't have to tell L's name as it seems quite feasible to arrange that Misa would see L's face and use her own eyes.
