Hi FasaSoftware,
I got interested in this topic and wanted to see if I can have a go at making a very basic file manager for Android. The result is the attached zip file. I included the apk file so you can test it out. Basically, it implements the idea of selecting files and navigating the Android file system.
Here's how it works:
On the list field, you tap or highlight the folder you want to visit. That folder is opened and you are shown what's inside (like an actual file picker). The folder path is shown on the small field on the lower right side. If you wish to go up one folder level, you press the Go Up button. The "Open" button will only enable if you have selected a file instead of a folder. When you press the Open button, it will show via alert box the path to your selected file.
This was a very interesting challenge and it gave me more knowledge for LiveCode. Hopefully it helps you out with the basic idea of implementing your own file picker.
It's too big to attach so I uploaded it elsewhere:
http://www.mediafire.com/download.php?rfzg2xgmdo4ntdm
I've tested it on a rooted ICS 4.0.4 phone but I don't think that matters though.