Open a terminal with CTRL+ATL+T keys combo and
Bare minimum you need to issue the following command:
sudo usermod -a -G sudo USERNAME
Replace USERNAME with the username of the user you want to promote.
sudo will ask for the password of the already existing administrator.
On my particular system, I am a member of the following groups:
usermod -a -G adm,cdrom,sudo,dip,plugdev,lpadmin,sambashare USERNAME
Mostly a copy paste from a question I bountied on SuperUser here.
If you upvote this answer, please go upvote that one. It deserves it, as it is more correct than the more heavily upvoted and accepted answer.
Login from the user who is already administrator, then click on Dash (Ubuntu icon up left) write
user and click on
unlock by clicking the appropriate button and giving the administrator's password.
Last, click on the user you want to promote and change from Standard to Administrator.