I am designing a full featured website that includes a blog on the homepage. I want it to be full featured, but I dont want to spend forever coding it. Is there any good blogging system that can be easily integrated with CodeIgniter?
Honestly I would separate CodeIgniter from the blog code. If you look at many websites that have a blog these days, they will often have the main application at www.example.com and then host their blog at blog.example.com, where they will have an installation of Wordpress (which I recommend, since you already have a php/mysql stack) or something similar. You can still make this work if you'd prefer to have it at example.com/blog but depending on your site you may have to do some modification of things like your .htaccess file, as well as your blog system theme if you want it to look like the rest of your site.
Expression Engine 2.0 is built on CodeIgniter, and made by the same company Ellis Labs. It's pretty decent, and will be very familiar to you if you're already familiar with CI.
If you plan on making a public blogging platform, you can not use Expression Engine due to licensing violation. see: http://expressionengine.com/sales_faq/article/myspace_blogservice/
For a free alternative, try http://pyrocms.com/ . I have been thinking about implementing myself as I wish to implement a blog on my current site.