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Medical Mystery - Introduction
Chances are that you'll pick up this quest when you do the Restoring Hope quest, although you don't have to do it then. You should be able to pick this quest up from Dr. Richards at any point. He spends most of his time in the medical tent and I believe that he's even prone to sleeping on one of the unoccupied beds (I've seen it happen at least).
Dr. Richards will mention that someone's been stealing the Hydra supply from his medical cabinet and he'd really like for you to figure out who's behind it.
You have a few options for this one. If you have some points in Medicine, you can take the medical route. Pickpockets can take care of this in a few moments and everyone else can settle in for a stakeout.
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Medical Mystery - Catching Private Stone
The stakeout is pretty easy. You can just wait outside, or even inside the medical tent until night. Dr. Richards should mention that the best time would be during the guard change at 2 AM. Wait until just shy of 2 AM and go into the medical tent. Private Stone should be sneaking around inside and working his way to the medical cabinet in back.
The game should register that you've seen him, which opens up the first end of the quest. You can just walk away and report the theft to Dr. Richards in the morning.
You can also talk to Private Stone right now. Since you've caught him red-handed, he's a little more open. You can make him hand over 5 hydras as a bribe for your silence. Note that you can also just turn around and report him to Dr. Richards anyway. You'll take a bit of a karma hit for both accepting the bribe and then betraying him, but you'll be well rewarded.
If you have Speech higher than 60, you can convince him to confess to the crime himself. This option is technically the best solution as far as karma and reputation.
If you have more than 25 points in Medicine, you can ask Dr. Richards about the symptoms of Hydra abuse (and convince him that you actually know enough about medicine to successfully diagnose someone, otherwise you assume Hydra makes them purple and close this option).
If you then have more than 50 points in Medicine, you'll be able to catch Private Stone just by inspecting him. He's usually leaning against one of the tents close to the Quartermaster. Look by the two bridges for a soldier that looks a little unique. You'll be able to diagnose his abuse and force him into the same position that you would if you would have done the stakeout version.
If you want, you should be able to pickpocket an empty Hydra syringe off of the Private. This will count as enough evidence to turn him in, just don't get caught picking his pocket.