I personally cannot see any near term replacement for what SQLite offers. In my world, SQLite became the way in which I now manipulate a scope of structured data on a single logical disk. I mostly view it as a very reliable and friendly layer on top of basic file IO which also happens to be compatible with virtually every computer on earth. I can copy a SQLite database out of an embedded microcontroller environment and directly open it on my x86_64 workstation, edit some values, and then copy it back down without any fears of compatibility issues.
* Simple to embed into you application (no setting up a separate database, &c).
* Fantastically backwards compatible
* Public domain (so easy to comply with the license)
* High quality: Reasonably fast, reasonably efficient, reasonable feature support.
The lack of any of these would add significant drag. It's difficult to imagine any other project catching up and displacing it at this point.