Displaying content based on drop-down menu selection sitewide

You need to set a cookie for this. You can then retrieve the cookie’s value on any page where you need it, or in the header.php.

There are several ways you can do this, you can set cookies either via php or JavaScript.

Here is a good example on how to set a cookie with php in WordPress. For JavaScript, there are several options. There’s a jQuery cookie plugin, I recommend to google around for the different options.