I have tested this and unfortunately not found this to work. (using Blender 2.78 and Godot 2.1)
Packed the texture(s) and made sure "File|external data | Automatically pack all into .Blend " was checked and checking the "Copy Images" checkbox in the BetterColladaExporter - and no textures are copied anywhere after the import of the Blender scene.
In fact, even if I save the .dae file AND the PNG texture to the same folder as the Godot .scn file (before doing the scene import) the texture is still NOT applied to the mesh, even though the texture is there in the same folder. Everything manual. :(
This gets to be quite a trial when there are dozens (or hundreds?) of textures that do not come over and need to be copied by hand and assigned individually.
Plus Godot has a weird habit of saying the source files have changed after you run your Godot program when the source files haven't actually changed, then it does an automatic update and you have to do all your texture assignments over again; as well as reassigning scripts, reconnecting node Signals etc.
If vonflyhighace2 or anyone has any suggestions, I am desperate for this texture import function to work.