ntrrobng

Oh missed some. Also change this:
<a border="0" href="
to this:
<a border="0" target="_blank" href="

Sorry this is horribly ugly.

If you want all of them to display in a new tab, look for the following in the same file:
<a href="
Replace with
<a target="_blank" href="

There will be more than one place that this occurs. If you only want it to open in a new tab if the alt text exists, it will take a bit more modifications…

To open a link in a new tab, you need to add a target to the link anchor in the HTML (see stackoverflow answer below), which is built in the rectangular_talavera and square_talavera functions.

https://stackoverflow.com/questions/6296013/how-can-i-open-a-link-in-new-tab-and-not-new-window

I couldn’t find a solution to this so I modified the jetpack plugin to use the image alt-text (if it exists) instead of the image URL. There’s probably a more elegant solution but I was in a hurry and it works for my purposes.

The file that I changed: ../wp-content/plugins/jetpack/modules/tiled-gallery/tiled-gallery.php

There are different functions for different gallery sizes (rectangular_talavera, square_talavera, etc) that all use the link, so I created a new function for the link logic:

private function get_link_url( $image ){
	$alt_text = get_post_meta($image->ID, '_wp_attachment_image_alt', true);
	if( $alt_text != ""){ // should probably check if it's a URL with regex...
		return $alt_text;
	}
	$link = $this->get_attachment_link( $image->ID, $orig_file );
	$search_text = "/project/";
	$project_pos = strpos($link, $search_text);
	if($project_pos !== FALSE){// this checks if the link is already a project and truncates the URL at the project itself instead of the project picture, since this is was the primary use case in my project
		return substr($link, 0, strpos($link, "/", $project_pos + strlen($search_text)) + 1);
	}
	return $link;
}

This needs to be used in the two functions where the link is inserted. I found it in two places: rectangular_talavera and square_talavera, where I changed
$link = $this->get_attachment_link( $image->ID, $orig_file );
to
$link = $this->get_link_url($image);
which utilizes the new function above.

Voila, using the gallery image alt-text (or parent project) as the link instead of the image URL.