My Fucking Mail would be a better name. As in, it's mine, do fucking not read it. Sorry for the profanity but I think it fits how many people feel about this.
I do share your sentiment! The shorter "MyMail" makes the same point without needing the profanity, and your more emphatic name would then be available for the most desirable MyMail client...
I did think of that while posting, but edited it out of my comment as 1) too cheesy and 2) it doesn't really make the point, because it just sounds like your typical big-corp lame name.
It's a bit late for the HN thread now, but how about MFM protocol, but pronounced as "My Mail" just as RTFM is sometimes pronounced as "Read The Manual."