Your input code formatting is wrong:
echo "<input type="radio" value='$id'</input>";
should be...
echo "<input type='radio' value='$id' name='staff'> $name<br />";
You need to either make all of the quotation symbols within your double quotes single quotes OR escape the double quotes inside the echo with \" as such:
echo "<input type=\"radio\" value=\"$id\" name=\"staff\"> $name<br />";
I also added a name attribute because for radio buttons to work properly they all need to share the same name.