Step 1: Talk to Old Lady Gibson near Novac and ask about Helios One. This begins the quest.
Step 2: Wait or spend 9 days in game. (You'd have to wait 24 hours 9 times, sometimes it takes a little more) then go to HELIOS One and talk to Ignacio. Talk to him about ARCHIMEDES.
Step 3: Wait or spend 2 days in-game. You'll get a message from the Followers of the Apocalypse and/or Brotherhood of Steel (depending on if you've destroyed them) through ED-E. Bring ED-E to whichever faction you wish. Followers will give him an upgraded Laser, Brotherhood will give him somewhat better armor. Either way he is generally cleaned up.
Step 4: After you give ED-E to either faction, wait or spend 3 days in game.
Step 5: You will find him in Primm where you first repaired him.
Quest complete.