I'd argue that the name should work on the assumption that it is the defacto standard, and so there is no need to distinguish between "normal mail" and "new mail". It should also contain a possessive pronoun. (Something like "YourMail", if not already taken?).
With such a name, when someone attacks the mail service it will be difficult to avoid attacking the person, as an attacker will have to attack "YourMail". Attack the person and the person will react.
Not distinguishing between normal and new will have two effects:
1) It will seem a normal thing for the uninitiated to use it.
2) In the case of an attack, it will be easier for people to say the attack is directed at themselves, since the generic name will draw people in, rather than letting them dismiss it as an exclusive club.