"I used to give the the option of doing either MySQL or SQLite"

SQLite is good choise for absolute beginners.

Later, when teaching "real" multi-user RDBMSes, although MySQL may be more popular, it makes more sense to teach PostgreSQL as "default open source database". Both will do the job, but PostgreSQL has got more stuff right from the beginning, which is especially important when learning. Think PHP vs. Python, PHP may cut few corners but it's not ideal language for teaching generic concepts.