Before you can start to eliminate single points of failure in a system you need to identify all the components that make up that system.
Since your goal is to make the LAMP stack highly available you already know that you need to account for Linux, Apache, MySQL, and PHP. However, there are other less obvious components that need to be accounted for in the architecture. The options and combinations are almost limitless and architecting a well-thought-out solution can be challenging. Use this guide to get started.