How to kill a daemon with its name gracefully?

Your question is not clear, you talk about a daemon in the title, but in the body only talk about a generic process.

For a daemon there are specific means to stop it, for example in Debian you have

    service daemon-name stop

or

    /etc/init.d/daemon-name stop

Similar syntaxes exist for other initscript standards used in other distributions/OS.

To kill a non-daemon process, supposing it is in some way out of control, you can safely use killall or pkill, given that they use by default the SIGTERM (15) signal, and any decently written application should catch and gracefully exit on receiving this signal. Take into account that these utilities could kill more that one process, if there are many with the same name.

If that do not work, you can try SIGINT (2), then SIGHUP (1), and as a last resort SIGKILL (9). This last signal cannot be catched by the application, so that it cannot perform any clean-up. For this reason it should be avoided every time you can.

Both pkill and killall accept a signal parameter in the form -NAME, as in

pkill -INT process-name

On BSD-like and other distros, you will often have scripts in /etc/rc.d/ that typically manages starting, restarting and stopping daemons in your system. To stop a daemon you would either call the scripts with the absolute path e.g.:

# /etc/rc.d/acpid stop

or use the command:

# rc.d stop acpid

I highly recommend to try out this script for showing your systems started and stopped daemons:

#!/bin/bash

chk_status(){
  target=$1
  if [[ $target != "functions" && $target !=  "functions.d" ]]
  then
    if [[ -f "/var/run/daemons/$target" ]]
     then
       stat="\e[1;32m[RUNNING]"
     else
       stat="\e[1;31m[STOPPED]"
     fi

    printf "$stat \t\e[1;34m$target\e[0;0m\n"
  fi
}

daemons=($(for daemon in /etc/rc.d/*; do echo "${daemon#\/etc\/rc.d\/}"; done))

if [[ $1 != "" ]]
then
 chk_status $1
else
 for d in "${daemons[@]}"; do
   chk_status $d
 done | sort
fi