Don't get fixated on the classic resorts, or you'll miss out on decent options. Abkhazia is genuinely quieter, but there's a catch with documents and border control - it can be a real headache in September when there are still plenty of people around. With a seven-year-old kid, that makes the trip more complicated.
On the Russian coast, look toward Tuapse or the Sochi area, but not Sochi itself - places like Lazarevskoye or Khosta. The water in September is still decent enough for a child there, and there's definitely fewer people. The main thing is book your accommodation in advance; September isn't peak season, but good options get snapped up fast. Prices are already dropping compared to August, especially if you go later in the month. Anapa is also an option - cheaper there - but the water might already be a bit chilly, which could be uncomfortable for a seven-year-old if he's not particularly hardened to cold.
Also check out Gelendzhik away from the center, small settlements nearby like Betta. There's infrastructure, but tourist crowds are minimal. With a kid, that's the most convenient - not boring, but nowhere near as hectic as Kurortny Prospekt in Sochi.