I know Obi-Wan made a fool of Grievous in RotS, but Grievous did "win" those lightsabers from other Jedi based on the Clone Wars cartoons.
I see no reason a droid couldn't be outfitted with a functional lightsaber. The effectiveness of a droid wielding a melee weapon is questionable due to limited maneuverability, but simply keeping it on and swinging it back and forth doesn't require any particular aptitude with the Force (Han Solo, The Empire Strikes Back). IMO Jedis are effective with lightsabers because the Force allows their actions to flow with the moment and anticipate hostile actions. Not to mention that they are specifically trained to fight with them.
Anyway, I would train Mission.