I recommend a Quantum ProDrive LPS. These drives were used by Apple in the early 1990s, mostly in LCs, and can easily be had for a few dollars (often as part of an LC or LCII). The ProDrive LPS is quiet, quick, and has an easily accessible connector for an LED wire (which you'll want in an SE since it has a HDD indicator lamp in the front bezel, driven by a bulb in the SE's mounting bracket).
To access the SE's hard drive, you'll need to first remove the cover. If you don't already own one, buy a long-shafted Torx T-15 screwdriver. There are two screws near the ports and two underneath the case handle. Work the case apart with your hands (don't use a flat screwdriver or other tool on it or else you may damage the case). Once the back cover has been popped off, locate the hard drive. It's facing connector-side up. Disconnect the power cable (four-prong) and the SCSI cable (ribbon-style). (If the old drive is a MiniScribe, the power cable is a bit tough to remove due to its angle, but patience will pay off here--just keep wiggling it). There are two Philips head screws at the base of this bracket that connect it to the floppy bracket. Remove both, and make sure to use a relatively large screwdriver (since these screws are often held in very tightly, especially if the bracket has never been removed before).
Once you get the bracket out, disconnect the wire that connects the LED to the drive's front pins. Remove the four screws on the side and pop the new drive in. You may have to play around with the positioning until you find a suitable place for the screws. In some cases, you may only be able to get two screws in, but as long as they are holding the drive in securely and in line with the bracket (not tilted or at an angle) you should be fine. Connect the LED wire to the pins on the drive and reconnect the drive bracket. Pop the two cables in and your drive is good to go.
The CRT's big danger point is the anode cap, which is the red suction cup style connector. Most SEs self discharge their CRTs, but to err on the side of caution, make sure you don't touch it. (If you do need to remove it at any time, it shouldn't be an issue, but I always use an insulated pair of pliers to remove these cups). Also, when you pull the drive bracket out, be sure not to bump the CRT's neck. As long as you are careful and take it slowly you shouldn't have any problem with avoiding this fragile piece of your display.
Speaking of the LED, has anyone else tried different colors for their LEDs in SEs? I once put a blue LED in an SE's hard drive bracket and it looked awesome! (Most of them came with red or amber, although I did see a stock SE SuperDrive with green LED at a school in the early 1990s). If I can find violet, I want to try that next.