Writer's Fallout Shelter

It really depends if I'm gripped enough by the plot or the character herself. Names are irrelevent. If it helps you could always call her Jill for now or something equally balanced that doesn't cause you to shudder. Perhaps do a little freewriting getting to know her better. Characters names are their own and for the most part they aren't really important. Mary Sue has a tale to tell and she's chosen you to tell it. Since it's there I'd be writing about it. Never look a gift character in the mouth not matter what their name is because denying it will generally lead to a massive writers block.

I'd agree with your friend. Write the book. You can always change the name if you're sill objecting to it in the future but by the time you reach The End you'll probably have learnt that Mary Sue was exactly the name she was meant to have. Characters have a way of owning their names and no matter how often you try to call her Jill she'll still be Mary Sue. There is nothing wrong with that.

Have fun and keep writing,
Yours truly,
Rebecca Laffar-Smith