Good questions for a Harry Potter character quiz include "To whom was Remus Lupin married?" and "Who hands Dobby the house elf the sock that grants him his freedom?" The answers are Nymphadora Tonks and Lucius Malfoy. Another question about Remus Lupin is "Which of James Potter's best friends is a werewolf?"
Make an interesting Harry Potter character quiz by asking questions about characters other than Harry, Ron and Hermione. Ask "Which professor wrote a series of self-aggrandizing memoirs that turned out to be largely fictitious?" or "Which professor was found locked in a magical trunk, having been replaced by Barty Crouch?" The answers to those are Gilderoy Lockhart and Alastor Moody. Another good question is "Which professor's veiled prophecy said that Harry Potter had the power to defeat Voldemort?" That professor is Sybil Trelawney.
Hagrid is a popular character. Ask questions about him, such as "Who is the gamekeeper who later becomes a professor?" "Who keeps a three-headed dog named Fluffy?" or "Which character was sent to deliver Harry Potter to his first year at Hogwarts?"
Luna Lovegood is a well-known character, but her father's name results in a more challenging question. Ask "Which character publishes a wizarding tabloid called 'The Quibbler'?" True Harry Potter fans remember that his first name is Xenophilius.