batting (n.2) Look up batting at Dictionary.com
"action of striking with a bat," 1610s, verbal noun from bat (v.2). In cricket, from 1773. Baseball batting average is from 1867.
batting (n.1) Look up batting at Dictionary.com
"sheets of cotton fiber," 1875, variant of obsolete bat "felted mass of fur, wool, etc." (see bat (n.1)), on notion of "beaten" fabric.