Medical Mystery

Medical Mystery is a side quest in Fallout: New Vegas.

Detailed walkthrough
The easiest way to complete this quest is to simply wait, hidden, inside/outside the medical tent starting around 1:00 in the morning. Around 2:30, Private Stone will enter/exit, crouched, and attempt to steal the supplies. Depending on your Speech level, you have a number of options in dealing with the situation.

Pickpocket Private Stone and take the empty hydra syringe (the clue) from his inventory. After that, report him to Dr. Richards.

Alternately, if you have passed the Medicine check for Dr. Richards to tell you about the symptoms you can talk to Stone by passing another 50 Medicine check; this alleviates the hassle of waiting until 2 a.m. for him to sneak into the medical tent. (Stone can usually be found hanging around the barracks.)

Another option is to talk to Sexton about missing drug supplies; he'll tell you Stone has been absent from his bunk late at night. Confronting Stone will yield an option to make him confess with a 50 Speech check. You can then attempt the 60 Speech check to make him turn himself in for a total of 110 XP.

Another way to complete the quest is to get Private Stone to promise to stop stealing the supplies through a Speech check. When you return to Dr. Richards, you can tell him that you could not find the missing supplies and you will complete the quest, but you will receive no XP nor NCR fame.

To gain more XP you need to speak to Dr. Richards about the missing supplies, pass the Medicine check (50) then go and talk to Stone about the medical supplies and again pass the Medicine check (50). Tell him about the harm his actions cause to soldiers that need the supplies, then say he should turn himself in and pass a Speech check (60). After that go and talk to Dr. Richards and he will tell you that Stone turned himself in and then you will gain 300 XP, good Karma, and NCR fame.

Bugs

 * Private Stone may disappear, leaving you unable to complete this quest. The quest marker will point to where he usually hangs out (outside the Command Center), but he's not there. It's possible he's trapped under the terrain. He never appears in the medical tent if this happens.
 * Waiting hidden inside the medical tent between 1:00am and 2:30am can fix this.
 * Reloading to a point before you entered the camp (the first time) and reentering may fix this problem.
 * This is now fixed in official patch 1.06. The pointer will still point to the empty spot, but you can now find him either inside or outside the medical tent at the designated times.
 * After the completion of the quest with turning Stone in, he may be absent from the jail and be wandering around still (and stealing the hydra), and all he will say is "Get lost."