How can I add a single image from a gallery into the page header?

You can use get_post_galleries to return an array of all the galleries in a post. The data returned includes all the image URLs and IDs, so take your pick of those to suit your case.

$galleries = get_post_galleries( 
    '', // assume current post, if not put a post ID here
    false, // return an array of data rather than the gallery HTML
);

$images = explode(",",$galleries[0][ids]);
// use the first gallery.  Traversing the array to collate all
// instances of [ids] is left as an exercise for the reader

$index = array_rand( $images );
$imageID = $images[$index];

// then do what you like with the image ID here

get_post_galleries takes two parameters:

• the customary optional post ID so you can either use it inside a loop or query any post.

• a boolean to indicate whether you want the whole gallery HTML (if true) or arrays of image IDs and URLs. The URLs are those of the image size specified for the gallery shortcode.