"The wand chooses the wizard". False. Some of the most powerful wizards don't even use wands. Everyone knows it's the name that chooses the wizard.
When a wizard comes of age, a name swells up from the murky depths of magical infinity and wraps itself around them. Names are released only upon the wizard's death, ensuring the names of the dead are passed on for time immemorial.
This can make personal branding a bit difficult, especially if you get something like "Wizard Council Museum Clerk", "Grombolenus the Gross", or "Neebus".
While not every name is as catchy as "Evil Wizard", "Chris the Speller", "Volcano Wizard", or "Tim", your wizard name is still very important part of being a wizard, and thus you should work hard to make it your own.
It did choose you, after all.