Drive letter won't change if there is an issue with the drive seeing the disks. What is the issue exactly? Do the disks spin up but the drive shows as being empty? Does it make noises when trying to read a disk other than the smooth spinning noise? You could have just a hardware failure, I'd get a new drive first.
Well, you have tried several discs i take it, so you've ruled out a bad disc. Check to make sure the drive is enabled in BIOS and is showing up there first. There are certain windows issues that can cause a drive to stop working. You can try and boot the laptop from a disc (windows, knoppix, whatever) and if it boots successfully, the drive is fine and you need to look torwards the windows end of it. If the machine can't even boot from a CD (assuming it's not some ancient machine that doesnt support that) you can first try cleaning the drive. A cleaning CD or a few puffs from a can of compressed air can sometimes work wonders. If not, replace the drive. Don't use a compressor on the drives. The higher air pressure, and the tendency for air compressors to condense moisture in their lines, isn't good for cleaning optical drives. If it won't read a disc no matter what you do (even booting from a disc) then just replace the drive. They're cheap enough these days that you shouldn't waste too much time investigating it.