Medical Mystery

 is a quest in Fallout: New Vegas.

Quick 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 2:00 or later, 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.

You can also 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 around 2am 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.

Detailed walkthrough
When you actually find Private Stone trying to sneak his way out, you have several choices:

1) Ignore his movements and let him escape, then report to Dr. Richards. You'll get 300 XP and NCR Fame as a reward.

2) Tell Stone to turn himself in. This requires passing a Speech 60 challenge. You'll get 60 XP plus an additional 300 XP, gain Karma, and NCR fame reward from talking to Dr. Richards after Stone turns himself in.

3) Confront him and say you'll let him go if he can make "worth your while". Stone will give you five Hydras for this.

4) You can also turn Stone in even if he gives you the five Hydras. You'll get 300 XP, NCR Fame and a Karma loss for betraying Stone.

5) Talk to Stone in the medical tent, let him escape, and then turn him in to Dr. Richards. You'll get the normal 300 XP and NCR fame reward.

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 Centre), but he's not there.  It's possible he's trapped 'underground'. He never appears in the medical tent if this happens.
 * Reloading to a point before you entered the camp(the first time)and reentering may fix this problem.
 * You can also use the console command.