I'm not working (still student), so it's just going to be about what I heard from others and I don't live in the same country and don't know the law where you are but:

• To me, the average pay alone is one of the worst arguments once you've started working and can talk about your own achievements, your situation,... To make an average, you need people below it (many, usually a bit less than a half of those taken into account), saying that you deserve to be paid above that many people won't help you if you can't back it up with results/examples.
• The boss knows you're unhappy with the pay, that's great, but does he know why ? You're the only IT in the school ? Try to ask for when you could have a real raise (3%, yay, I can keep buying the same stuff that increased by 3.% ...), if positions might open with (maybe) more responsibilities & higher salary
• You don't have to leave, but you probably should look for others job.
  • If you don't receive positive answers, you lose nothing
  • If people/companies want you, you have an option to either negotiate a pay raise (you'll have a clear reference of your worth), or just leave.
  • Either way, make what you want clear. From what I can read, you'd rather stay where you are, and that's fine.
• If you feel like you're not being properly represented, you might be able to ask to talk higher, I don't know how things are done in your area. If not, you might be able to go ask around if "everything works fine" on a regular basis: sometimes not getting a raise is just due to the fact that people don't know you ; give them the occasion to know at least your face if not your name.
• No matter what, don't claim you might leave (if you don't get a raise) if you don't really have a backup plan ready. Bluff will only put you in a corner.
• Depending on the attitude of those around you, you can still say that you're thinking about it, that might help and would make things clear (tell them why you're thinking about it though)