After a self-imposed shutdown during Luke’s absence, R2-D2 again proves his importance. He partners with BB-8 at the Resistance base to complete the map to Ahch-To. That sends Rey and Chewbacca on their way to seek Luke’s assistance. Luke is initially reluctant (understatement) to help.
It’s only when the sassy little droid shames Luke by showing him the original plea from Leia, begging Obi-Wan Kenobi for aid, that he finally acquiesces. Luke joins the fight, providing Leia and the others the time they need to escape the First Order, seeming to give his life in the process.
So fine, I’ll grant you this: Artoo makes an impressive case for most influential character in Star Wars, indeed.