When writing a book of your own, you absolutely can use other books on the topic as references. And you can quote those other books. In both cases, you want to credit the other book.
For quotes you put a citation directly into the text, or put an endnote or footnote and put the citation in there (or a combo of both).
For reference materials, most authors of nonfiction will include a bibliography at the end. You can also cite books in the text or in foot/endnotes if you are rephrasing something of theirs.
If your goal is to use this book as practice for your own writing, then, sure, go ahead. Henry has some great tips on how to do that effectively. But if you'd just like to re-write the book into a version that works better for your theme, the answer is no. Sorry, that crosses the line into plagiarism. Even if you give full credit to the other author and publisher, it would be republishing their work without permission. There's a limit to how much of someone else's work you can use.
Here's the thing though, if you want to write a history, you need more than one reference. Don't use just one history book: use a dozen. Use articles and websites and some original research if you can.
Because you are only interested in a subset of video games, you especially need a large number of sources. Most of the popular video games will be ones that don't meet your criteria and most of the books will focus on the popular games. Many of those popular games will be family-friendly of course. Depending on how you define a "Christian" game, the ones you talk about may be lesser known. In order to find the more obscure games, you'll need to scour those sources.
If you want to write about the history specifically of development of the games you include, your best bet may be to find the creators and interview them. The more general history of video games is easy to find already written, but who are the creators and producers of the games you want to promote? How did they get their ideas and get the games made commercially? Those answers might not be in existing books, at least not collected altogether.
If your sole goal is to re-write an existing historical account to meet your needs, that won't fly. You can do it for personal use, but not to publish (putting it on a website available to the public counts as publishing).