The Badge of Vector Prime was a symbol of Vector Prime's power.
The Badge of Vector Prime was worn by Vector during the Age of Primes. It is unknown whether the badge had powers of its own, but Alpha Trion recalled seeing it often adorning his brother's armor.
During the Age of Rust, Alpha Trion journeyed to a node of Vector Sigma to try and find some means of rekindling the spirits of Cybertron's population. He hoped for the Matrix of Leadership itself, but Vector Sigma could not bestow that gift upon him. Instead, Vector Sigma turned over the Badge of Vector Prime to Alpha Trion.
Alpha Trion gave the Badge to Sentinel Zeta Prime, in the hopes that it would spurn the elected Prime into activity. Although he wore the artifact, it seemingly did nothing to change Sentinel's ways, and may have even acted as a confirmation for Sentinel to stay the course.