How can I physically identify a single drive in a RAID array?

Disk Utility (sitting in System -> Administration) will give you the serial numbers for all your disks.

Here's what I see (look at the top-right for the serial). You'll notice that this drive is within a mdadm RAID array. Disk Utility can penetrate the array for raw disk access.

I have 6 of the same model of disk in my PC so I drew a little diagram showing their position in the case and the serial number so I can locate them quickly on serial in an emergency.

The opposite is also true in that if a disk dies, I just need to find which disks are showing up and I can eliminate them until I know which serial is missing.

Edit: I'm trying to improve my bash-fu so I wrote this command line version to just give you a list of disk serial numbers that are current in your machine. fdisk may chuck out some errors but that doesn't taint the list:

for disk in `sudo fdisk -l | grep -Eo '(/dev/[sh]d[a-z]):' | sed -E 's/://'`;
do
    sudo hdparm -i $disk | grep -Eo 'SerialNo=.*' | sed -E 's/SerialNo=//';
done

(And you can crumble that into one line if you need to - I've broken it up for readability)

Edit 2: ls /dev/disk/by-id/ is somewhat easier ;)

If you have trouble matching the drive serial number or port indication with your disks' spatial locations, you can run cat /dev/sdz >/dev/null (where sdz is the failed drive) and locate the drive by its LED (or by ear if you aren't in a noisy server room). If the drive won't even power up, that should be enough to tell which one it is. Be sure to put a visible label on the disks for next time.

The info that udisks gives (either on the commandline or in the GNOME Disk Utility) includes the disk serial number. On the disks I have, the serial number is printed on the upper side and on the front side (the one on the other side of the one that contains the connectors), both as numbers and with a barcode. Unfortunately, most PC cases make it impossible to read those serials without pulling the disk out...

You can also find the serial numbers in /dev/disk/by-id/.

As your disk is off-line, I assume it isn't "seen" by the kernel currently? In that case, you might have to go by elimination: you want the disk with a serial number that is not listed...