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 at ~1:00 in the morning. Around 2:00 or later, a private 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 empty Hydra (the clue) from his inventory. After that, report him to Dr. Richards.

You can also have 50 in Medicine and ask Dr. Richards about the symptoms of Hydra. After that, confront Private Stone telling him he has the symptoms.

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.