Possible to dump AJAX content from webpage?

You can use WWW::Mechanize or another Crawler for this. Web::Scraper might also be a good idea.

use Web::Scraper;
use URI;
use Data::Dump;

# First, create your scraper block
my $scraper = scraper {
    # grab the text nodes from all elements with class type_firstname (that way you could also classify them by type)
    process ".type_firstname", "list[]" => 'TEXT';
};

my @names;
foreach my $page ( 1 .. 146) {
  # Fetch the page (add page number param)
  my $res = $scraper->scrape( URI->new("http://www.familiestyrelsen.dk/samliv/navne/soeginavnelister/godkendtefornavne/drengenavne/?tx_lfnamelists_pi2[gotopage]=" . $page) );
  # add them to our list of names
  push @names, $_ for @{ $res->{list} };
}

dd \@names;

It will give you a very long list with all the names. Running it may take some time. Try with 1..1 first.