The cocoa in chocolate chip cookies contains theobromine, which may mildly inhibit uric acid production 1. A single small cookie occasionally is unlikely to trigger a flare for most people.
Bad for you
Each cookie typically contains 9–12g of added sugar, which is about 50% fructose and directly increases uric acid production 2. Refined white flour also spikes blood sugar, potentially worsening hyperuricemia 2.