I'd go with Milly or
Lissa.
Mia isn't bad either. I think
Ellie and
Ellis are too contrived to come from
Melissa (this coming from a person who decided a few years ago that
Erin would be a good nickname for
Catherine, but never mind), and
Elissa is practically the same name.
I'm not sure how you'd tell people outright that you want to be called a different name, but I've found that by correcting people if they call you by a different nickname (for example, people call me Cath or
Cathy. I tend to make a face because I hate the nicknames, and I tell them that I prefer
Cate) and also signing notes/texts with your nickname helps it get it into people's heads.
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~~~~~ - Sparkleflower, Moonrain, Honeybud, Twinklesun and Sugarberry.