late 14c., "fit, suitable, proper; affording accommodation; opportune, favorable," from Latin convenientem (nominative conveniens), present participle of convenire "to come together, meet together, assemble; unite, join, combine; agree with, accord; be suitable or proper (to)," from assimilated form of com "with, together" (see con-) + venire "to come" (from a suffixed form of PIE root *gwa- "to go, come").
Meaning "personally suitable to ease of action or performance" is from late 15c. Sense of "at hand, easily accessible" (1849) is marked in OED as "Ireland and U.S."