I've had good luck using sqlite3 for both a hobby project (a photo organizer http://github.com/kghose/Chhobi) as well as for work (neural data analysis)
I've had good luck with SQlite as well, until I have multiple writers, or even concurrent readers or writers.
What's the use case? Are you looking for mobile platform compatible database, or for web backends?
MongoDB is 'lightweight' in the sense it's easy to setup and persistence of both simple and complex objects is a breeze. The query interface is a bit ugly but it may be worth a look.
It depends on what you need. sqlite if you need SQL queries /indexes,etc, try leveldb if key/value store will work, its fast and micro.