The problem is that you aren’t correctly escaping the input string, try:
echo "\"member\":\"time\"" | grep -e "member\""
Alternatively, you can use unescaped double quotes within single quotes:
echo '"member":"time"' | grep -e 'member"'
It’s a matter of preference which you find clearer, although the second approach prevents you from nesting your command within another set of single quotes (e.g. ssh 'cmd').