Here is a guide to flash the Intel Edison with the DFU mode.
Download the firmware from Intel
Install DFU util (USB Device Firmware Upgrade)
sudo apt-get install dfu-util
The following instruction can also be found in the Intel documentation but they forgot to add the sudo command (and without that it simply does not work).
~/Downloads/edison-image-ww18-15$ sudo ./flashall.sh
Using U-Boot target: edison-blankcdc
Now waiting for dfu device 8087:0a99
Please plug and reboot the board
Flashing IFWI
##################################################] finished!
##################################################] finished!
Flashing U-Boot
##################################################] finished!
Flashing U-Boot Environment
##################################################] finished!
Flashing U-Boot Environment Backup
##################################################] finished!
Rebooting to apply partition changes
Now waiting for dfu device 8087:0a99
Flashing boot partition (kernel)
##################################################] finished!
Flashing rootfs, (it can take up to 5 minutes... Please be patient)
##################################################] finished!
Rebooting
U-boot & Kernel System Flash Success...
Your board needs to reboot to complete the flashing procedure, please do not unplug it for 2 minutes.
My Edison drive was not showing up on my laptop. After flashing and rebooting again I got a fresh Edison which does show up as a drive letter again.