"Sorry, grandma, I know you've been sort of attached to your name for the last 80 years, but the white folks find it inconvenient for their computer systems. Don't worry, they promise they'll make something close for you."
Many of the clients of my ex-day job are married to legacy encodings like Shift-JIS precisely because they do think that their customers and students have a "right" to having their names written correctly. (Most of them also make a total hash out of foreigner's names, which I spent a good deal of time correcting. As far as I know my office probably still uses my name as test data, since it screws up about 80% of the systems we had, and it was cheaper to work around or patch than it was to fire me.)
The best solution is suggested in the other link: apologise for the technical limitations and offer a workaround. Don't demand that the world breaks solely to fit into your technical limitation.
If your name doesn't fit into Unicode, get a nickname. Bonus points if your nickname fits into 7-bit ASCII.
Say you want to order a package from somewhere. How does getting a nickname help? How do you explain it to the post office? I know it sounds like nitpicking, and it probably is, until it affects you personally. I've had my share of "name-mangling" and my name does fit into Unicode (not into ASCII though).
1. I don't want to inconvenience people with "weird" names.
2. I don't want to burden application programmers too much.
Requiring everybody to have a simple ASCII name would be convenient for programmers, but would be a big hassle for people whose names don't meet those requirements. "Be in the Basic Multilingual Plane or get a nickname" is a policy that, I think, provides a reasonable balance. Of course, supporting all of unicode isn't really that much harder, so I think that's a better balance.
I'd be very careful telling people what "rights" they have around their names.
EDIT: you can't really control what other people call you but you certainly can control what you call yourself.
Perhaps the actress from "Orlando" could start calling herself ~Swinton? Or maybe just "~"?