From Modern American

The name Carr appears to be a rare, modern, regional, or creatively spelled name, and a widely documented traditional meaning is not clearly established. For names like Carr, the safest interpretation is to treat the name as a distinctive contemporary form rather than assign it an unsupported ancient origin. If the name resembles a known root name or family name, its meaning may be connected to that root, but the exact meaning should be considered uncertain unless supported by family, cultural, or linguistic context.