Dating back to at least 1398, Grima is most likely a truncated form of a longer surname such as Grimaldi.
It comes from the Sicilian name Grimo (feminine: Grima) from the Germanic prefix Grim- meaning "harsh, ruthless". Via the Italian nickname grimm, it may share the same Germanic root as the English word grim, meaning "stern, severe". Kerima is a common Arabic name for women and means "generous".
Crossed swords pointing up in heraldry indicate a readiness to fight and protect.