In my experience the most important factor in justifying why you should be compensated better is to show that what you are currently being paid is below your value to the company. It's also best to do this within the normal review/compensation cycle at your company. "Out-of-band" raises are difficult to arrange and may actually cost you more in the long run.
Things like "I need more money" or "the cost of living is up" don't really hold much weight at all. The company is not a "charity" but rather in the business of making a profit. Overpaying for human talent is not a good way to do that.
Put monetary values on your accomplishments over that last review period and focus on those areas where you went above and beyond. That's what provide that convincing evidence for those above your manager who are allocating the usually scarce merit increase money. If you have big impact you will be able to justify a bigger increase. If you have a low impact, you will not have a good case.
Your problem is stated in your question:
Suppose that I, an average software developer with a satisfactory
record and an average skill-set...
Why would ANY company give you anything but an "average" increase if anything at all? You've defined yourself as AVERAGE! You're a commodity!
Change your outlook and the way you work and be exceptional instead of average. If you don't know how to do that, look around you at your coworkers. Who among them do you consider exceptional? Start there, watch what they do and how they do it. Ask for advice and even mentoring. Make it a point to practice being exceptional consistently. Be the first one to take on a new challenge. Develop a reputation for solving problems. Get known by manager other than your own. Be a "go to" person who is looked at as an expert in something important to the company across multiple teams.
Yes, that takes some time and a lot of work, perhaps even more than one review cycle, but it's worth it if you want to be more than "average". Average is easy, exceptional is hard.
I'll note that some people are satisfied to be average and that's fine for them. They are the ones who don't get the raise or the promotion and are often the first ones to go when staff reductions are needed. Nobody knows who they are and often don't really know what they bring to the table.