Here's a giant list of things you might want to consider taking to your dorm.

Essentials

• Hygienic products (toothbrush, soap, etc) to last you at least the first few days.  It will probably be easier to buy these things in bulk at the beginning of the year so that you don't have to make too many runs to the supermarket.
• Medication: If you know you need certain kinds of medication you should bring it.  While there are multiple pharmacies in Palo Alto, they may need a few days to get the drugs you need.
  
• Some form of government ID (you will not be allowed to pick up your dorm keys without this)
• Bedsheets, mattress pads (Stanford mattresses are of the "extra long" variety)
• Laundry basket / hamper
• Clothing (don't bring too much though.  Your dormitories do not have the biggest closets in the world, so you will need to moderate yourself.  Keep in mind that you can also swap out your closet when you go home for Thanksgiving or winter break)
• Bath-towel / bath-robe (you will probably be walking down the hall in just your towel after a shower)
• Backpack (or some kind of bag to carry your books)
  
• Technology
  
• Your computer (probably some kind of laptop)
• A lock for your computer so that people cannot just swipe it while your back is turned.  Most modern laptops are built with a metal slot, known as a "Kensington slot" that allows you to attach a special kind of lock that will let you lock down your laptop to your desk or bedframe.
  
• Cellphone (you can live without one, but it's very inconvenient to not have one)
  
• Surge protectors, extension cords (there are only a few outlets in each room)
• Phone cards (if you are an international student)
• Alarm clock (even if you think you are capable of waking up without one, you never know what might happen)
  

Useful Things

• Microwave (you might want to have hot cocoa on a cold night)
  
• Refrigerator (to keep fresh fruits from the dining hall overnight)
• Flashlight
• First aid kit (this should just be for occasional use, so you do not need something like a paramedic's rescue bag.  Each residence also has a peer health mentor who might have a supply of common drugs like Tylenol and Advil)

What NOT to bring

• Weapons, guns (it's illegal)
  
• Giant plasma screens (they will probably sink your productivity to unprecedented depths)
• Pets (not only are they are against school policy, but they are also a pain to take care of)
• Coins for laundry (your Stanford housing bill covers all laundry costs)