Just some things we noted about staying overnight at the Jeju Sauna:

• While i'd recommend leaving electronics at home as its the perfect place to just unplug, I also understand some people have to have their cell phone, computer, or DS with them depending on who you are, so do know that there are plugs around.  I'd suggest bringing an extension though, if you have multiple things to charge, as the availability of the plugs is kind of first come first serve. 
• That being said, there is free wifi should you bring your phone or laptop.
• Hair dryers, shampoo, and conditioner are provided, as are basic things like a razor, tooth brush, lotion, hair spray and 'towels.'  Tampons and bathing suits available for purchase.
• If you plan on sleeping there, bring ear plugs and a sleep mask.  The sleeping room fills up quickly so you may end up looking for a place to sleep within the common room or one of the saunas, and don't think the "Sleep Room" will provide you total peace and quiet.  Some people don't seem to care if their children run in and out between 2 and 3 am while shouting and playing.
• Mats are provided for sleeping, technically speaking only one per person, but i'd recommend bringing your own yoga pad or something for extra cushion.
• Bring a pillow as well, because the ones you use in the sauna are not going to be agreeable for overnight slumber.
• Budget for a massage if you're wanting one.  They're not outrageously priced, but they will catch you off guard if you're not prepared and are watching your money like moi. (Prices are on their website.)
• The fitness room is huge, with bicycles, treadmills, and elliptical provided.  There is a mirror wall along one side, and plenty of room to do Yoga or Pilates.  Bring your own mat and running shoes though.
• While I can't speak for other service providers, Sprint does not get a lot of bars in there, and my phone just seemed to eat up battery trying harder to GET a signal than actually having one.
• There is no special menu for breakfast.  What they serve is what they serve. 
• Lockers are provided, but don't think full body lockers.  These are your basic, gym locker room style lockers, if not just a smidgen smaller in the width department.

Basically, treat it as if you're going to go for a slumber party old school style, just short of rolling up your sleeping bag and onesie.  We brought along a little tote with stuff like clothes, sketchbook/books, hair brushes, etc, but small enough to fit into a locker.  I think my tote was one of those vinyl bags from The Fresh Market if that gives you any idea of size.