My theory is that Peridot's weapon are her fingers and her mind. She is basically the stereotypical nerd of the Home World gems. Jasper is a "bully" towards her because she is useless in battle without her "fancy tools". She is a technician for the Home World, sort of like the AV Club. Jasper is like the football jocks, she even wears a helmet. Honestly, Peridot is only a threat when not monitored. Her fingers can do many things like make a touch-screen operation device and make her fly like a helicopter.