Another possible option (simple, but less secure) is to enable setuid/setgid permission for nice executable.
sudo chmod +s /usr/bin/nice
setuid assign any user who can execute the file the effective UID of the file owner's (in this case root) when executing the file, thus is equivalent as running as that user. setguid does the same for effective GID.
Or you can do it more securely:
# create a system group named `nice'
groupadd -r nice
# set the owner group for `nice' executable
chgrp nice /usr/bin/nice
# disallow other users to run `nice`
chmod o-x /usr/bin/nice
# allow anyone who are able to execute the file gain a setuid as root
chmod u+s /usr/bin/nice
# add your user to the group
usermod -a -G nice <your-user-name>
Beware you need to re-login your account for new group to take effect.