I actually had the same question, based on Tarzan, as I am trying to writ e book called Greystoke & Man Cub. Mowgli is in public domain, as is Tarzan. However, Tarzan is trademarked. As far as I know how trademarks go, as long as you are not promoting the trademarked name on the cover or in ads, where people can see it, where the name used is being promoted for commercial purposes, then you are fine. At least, that it what i was told by various people.
In other words, I am also working on a game of public domain characters, but I think one or two in the game are trademarked. As long as they are sealed within the box, without any glimpse that they are in there (encouraging people to buy it), then you are fine. Also, the use of other verbiage should work too, as I plan to use "Lord John Greystoke" and not "Tarzan" or "Lord of the Apes" in my novel (or even "Jane") as all those are trademarked. In fact, I plan to keep his whole jungle experience out of the book, with very few references to his past.
ESB, Inc is pretty intense when it comes to trademarking the brand name of Tarzan. You should go and look at the list they have of items that cannot use the name. I mean, they sued a vacuum cleaner that took the name! Pretty nuts. So, always play it safe and use other terms which would not get you in trouble. Again, I do not want to give false advice, but this has been what's told to me on the matter. They also say to get an attorney to look at what you wrote as well, to see if it is legit. Sadly, that is usually the majority of what people have said, totally avoiding the question I was asking; which started to get annoying, is if they (including even some attorneys) not knowing the answer. Hope that kind of helps...and if anyone else can give better advice on the subject (or correct me), I am eager to know as well.