var=${var,,}

see man bash :)

#!/bin/sh

if [ -n "$1" ]; then
  let count=0
  while [ "$1" ]; do
    f="$1"
    shift
    if [ -f "$f" ]; then
      lc=${f,,}
      if [ "$lc" != "$f" ]; then
        mv "$f" "$lc"
        ((count++))
      fi
    fi
  done
  echo "Result: $count files have been changed."
else
  echo "Usage: lowerit file(s)"
fi