I have an existing WordPress website, and I would like to add my existing blog posts to a /blog subdirectory. What is the best way keep my old site home page and navigation, while adding my blog posts to a /blog subdirectory?

Try adding a category called 'blog' and assign the posts to that category. Then update your permalink structure to /%category%/%postname %/

Well do what has suggested but that would make the url look like http://yourdomain.com/category/blog/post-name/ So use some plugin or even SEO plugin by YOAST has this option where you can remove the /category/ from url slug resulting in the url structure you wanted. I'd also suggest that instead of placing all posts in Blog category,make all existing category a subcategory of this newly created category called blog.That way all your blog post will retain there existing categorization and you will get url like http://sitedomain.com/blog/category-name/post-name/

I think you can built a multiple theme wordperss . it's very reasonable. you can refer to: http://codex.wordpress.org/Installing_Multiple_Blogs

You can use the Export/Import function of WordPress (Under "Tools") to move your content from one installation of WordPress to another.  However, Viktor's suggestion is actually a better plan.

Are these existing blog posts on a separate WordPress site? If so, create a page called blog on the existing WP site. The go to settings > reading > and choose Front page shows a static page. Then choose your current home page for the site home page and the new blog page for the posts page. Export your posts from the old blog, then import them into your existing WP site.

mentioned the best way to do it. You basically are creating a blog page. The easiest way is to make a page and set the template to blog.