I'd suggest you to try with the FTP profile. This should allow you to transfer files. Check section 6.13 of the Intel® Edison Bluetooth® User Guide for more info about it. I'd also suggest you to check the following documents about Obex:
Thank you for the detailed response. However, the methods delineated in the resources you posted seem to assume that the reader is doing a manual transfer of files. The only feedback the Linux would need from the phone is commands, while the phone expects a continuous stream of integers from the Linux - so I'm not sure if FTP/OBEX would work for me.
Sorry for the delay in my response.
I suppose that you paired and connected the Edison to your Android device. An from your original post I also assume that you are making your Edison discoverable. However, did you set the Edison to trust your Android device? I've faced issues similar to this, where my phone won't connect to my Edison and it was because it was not set to trust the phone. Did you try that?