I often find the best weaknesses are the ones you can implement yourself, rather than relying on an opponent to find them out. My jedi master Jonyna's saber, for example, is more or less bond to her family line. As such, the crystal inside is haunted by her mother's spirit, who acts much like a helicopter parent. Should Jonyna stray from the path of good, become nothing more than a predator, that spirit can shut her saber off and make the crystal unable to be used by Jonyna.
Find something that makes your weapon unique to your story.