Here is the basic theory of negotiation and specific advise for your situation:
Theory: in a negotiation both parties are trying to get a deal that is better than the Best Alternative To A Negotiated Agreement (BATNA). So what you’re trying to do is say if this negotiation fails what am I going to do instead. Theoretically this is where your leverage always comes from because a good BATNA enables you to threaten to walk away.
Practical: in your post I understand what you want, but I don’t understand your BATNA. If your friend refuses, or worse let’s say he fires you right now, what would you do/make instead?
The easiest way to get a win-win is the following:
1) go find other gig offers - compare the price others are willing to pay you (recognizing there’s some risk the next client is not as friendly/easy to work with)
2) Let’s say you get 3 offers at $15,$16,$17. Show these offers openly to your friend and say “look I like working with you and want to continue but the market rate for my work is much higher than what I make right now”. This is actually a win-win because if you’ve done your research right, these are the same offers your friend would have to make to replace you. Now the conversation isn’t about what you want vs what he wants, but what each of you would do if you can’t get to an agreement.
3) Make your offer (say $18) and explain why it is higher/lower than the market avg (you can trust me, you know I do good work, etc).
4) Truly be willing to walk away if it doesn’t work. Not everyone is rational, not everyone will believe your alternative, not everyone wants a quick resolution. Unfortunately this is the hard part with friends, but if it doesn’t work you just must be willing to walk away (try to keep it isolated to your business partnership and not your personal friendship).
Notice, throughout, the premise is: here’s what my alternative is. Without this, you’re just hoping for him to take action out of kindness. It’s certainly possible, but it sounds like you are confident in your skills and value and ready to move past charity to career.
Good luck. Ask questions here if you have them. Let us know what happens.