I have a suitable page, archives.php, located in the theme's root directory, but I cannot get it to show up as a page template. Is there something else I need to do so that it is recognized?
Archives.php is usually not a page template - it is a file used to display all the archived posts (usually some sort of list ...) It is usually triggered by a function that deals with archived posts like
A page template file has to have this code in the begining ;
in order to show as page template ..
so should you really want to make archives.php to work as a template file -
if you have done correct, it would appear so there is something you are missing. All you need to do is make sure your archive page template resides in your theme with proper Template Name: Archives once done. Go to your pages add new page on right rail it has to appear in drop down.
For your reference have a look at