After you clear the sand from the EDIE, you can upgrade using XP, right? Right.
I went into MECHANICAL >> WORK BENCHES and clicked the Weapon Workbench, then I bought the other workbenches. Now the Weapon Workbench is unlocked!
I'm not certain which action triggered it, but I am very happy right now :D
EDIT: You can find the broom above EDIE. Just equip it and use it as a weapon on EDIE to sweep the sand off of it. Press 'Q' to enter the upgrade menu and follow the steps I outlined previously.
