Help, my robot is naked!

Your ship's droid companion uses standard droid parts (produced by Cybertechs, bought for commendations, or found as quest rewards), but no parts will help him succeed at missions - affection is the only thing that can affect missions.

Just use him for gathering missions, and phase him out as you get the rest of your companions — anything he can do, your other companions can do better.

As Raven Dreamer noted, you need droid parts to make him companion ready. And why would you use this incredibly annoying heap of scrap on your missions? Because he's the only healer pet you get at early levels unless you are the bounty hunter. You can equip him on the cheap if you always select the droid parts on the companion gear rewards for the first few you do. (Companion gear normally shows up as the rewards for the any of the non-main story line, planetary or class)

The companion system is set up so you will mostly have one less deployable slot than you have companions. This means that you should always have a free companion to run around with, although if you spend a lot of time in a group that won't matter much. Which puts the question of which companion should be allowed to see the world while you work the others like little third world children in your own personal sweat shop? Since affection controls both the time it takes to run a crew skill mission AND how often you get a critical success on crafting, any hardcore crafter should outfit the droid and max out the affection of the others ASAP! Assuming you put in 160 hours getting to fifty having your maxed out affection nets you around 43 "bonus hours" on your crew skills... which means more purples, materials and every thing else that you use to make MONEY. (The math there is based on the Bounty hunter companion gains... I know the trooper gets even more as they get their companions earlier than the bounty hunter does.)

If you aren't a crafter? That droid sucks... kick his crappy ass to the curb and only think about him when your cursing the fact the Bioware chose to put him right at the door to your ship so you have to listen to his sniveling voice every time you come aboard.