Change button text after ajax db update

You only need to set the text of your A tag using $a.text('abc') in your jQuery part. I did similar with the class in jQuery part of How to securely add an Ajax button to an admin page

jQuery('#content').on('click', 'a.rml_bttn', function(e) { 
    e.preventDefault();
    var $a = jQuery(this);

    var rml_post_id = $a.data( 'id' );

    jQuery.ajax({
        url : rml_obj.ajax_url,
        type : 'post',
        data : {
            action : 'read_me_later',
            security : rml_obj.check_nonce,
            post_id : rml_post_id
        },
        success : function( response ) {
            jQuery('.rml_contents').html(response);
            $a.text('Changed');
        }
    });

    //jQuery(this).hide();
}); 

});