Good question. A natural paraphrase of "I didn’t think anything of it" is: I didn’t pay attention to it. It does not mean that something negative came to mind. It means the person didn’t consider it important or didn’t see it as a big deal.

It's something like saying, "no offense taken" or, "it wasn't a problem for me." For example: Person 1: "Sorry, I didn't mean to interrupt you." Person 2: "It's okay. I didn't think anything of it."