The industry is dominated by Java, PHP and the usual client-side HTML, CSS and JavaScript.
Java can be used for a lot of things such as desktop applications, web applications (Applets, JSP, Java Server Faces, etc.)
PHP is a server-side scripting language usually used to connect to various databases on websites (Facebook uses this, and about 75% of the internet’s websites use PHP).
HTML is a markup language, that’s the foundation of web design, which goes hand-in-hand with CSS.
JavaScript is a client-side language and is used to tweak some of the functionality on the websites (drop-down menus, moving objects, etc. jQuery is a nice framework to use in this case.)
C++ is a lot more advanced and shouldn’t be someone’s first programming language, in my opinion.
C# is also very popular with Windows applications and is very similar to Java.
My personal preference is Java, not because it dominates the market, but because it is very flexible in the sense that it can be used across all platforms, i.e. all operating systems without modifying the code or importing any specific libraries. It works on the web just as good as PHP (and can also be used instead of PHP), and it is very easy to get into when it comes to desktop applications. Apart from that, there’s a ton of resources (which are free, such as textbooks and tutorials) your child can learn and study from.