As a memo to myself, and in case others aren't aware of this:
If you move the entirety of a mysql server (e.g., all databases, especially the "mysql" database) to a new Debian-based (Debian, Ubuntu, etc.) server, you need to make sure the debian-sys-maint user is created or updated.
If moving from a non-Debian-ish environment, try:
GRANT ALL PRIVILEGES ON *.* TO 'debian-sys-maint'@'localhost' IDENTIFIED BY '--password--' WITH GRANT OPTION;
where "--password--" comes from /etc/mysql/debian.cnf.
If moving from another Debian-ish environment, copy the password from /etc/mysql/debian.cnf on the old server to the same file on the new server.