Yes, you can. Normally bibledit stores its data in subdirectory .bibledit of the user's home directory.
Changing that is not done from within bibledit, but involves some manual work. This is done for the protection of your valuable data. To change the data directory to a new location, do the following:
Quit bibledit.
Open a terminal.
Type command
cd
to go to your home directory. Let's say you prefer to store the data on /mnt/server/.bibledit
Type command
mv .bibledit /mnt/server/.bibledit
to preserve and move existing data. (When there is no existing data, this step is not needed.)
Type command
touch .bibledit-datafolder-pointer
to make a new file in the home directory, that will point to the new data directory.
Open this hidden file in a text editor and put only this one line in it:
/mnt/server/.bibledit
That's it.
Removing file .bibledit-datafolder-pointer will cause bibledit to use the standard datafolder again.
See also backup.