client (n.) Look up client at Dictionary.com
late 14c., from Anglo-French clyent (c.1300), from Latin clientem (nominative cliens) "follower, retainer," perhaps a variant of present participle of cluere "listen, follow, obey" (see listen); or, more likely, from clinare "to incline, bend," from suffixed form of PIE root *klei- "to lean" (see lean (v.)).

The ground sense apparently is of one who leans on another for protection. In ancient Rome, a plebian under protection of a patrician (called patronus in this relationship; see patron); in English originally "a lawyer's customer," by c.1600 extended to any customer.