I want to ask user to select a file to open from external storage and receive it's path. Preferably I'd like to avoid excesive coding and use some standard method (well known, system-provided intent or similar). Said file is to be SpatiaLite db file (*.sqlite), so it won't show in Gallery.
Unfortunately there is no Android native file picker, but you can get an open-source one to add in your project, then modify it to fit your needs. Check out this SO question for a good start.