Combine 2 codes

http://codex.wordpress.org/Function_Reference/the_title

to get the title as a string to use in php, use

the_title('','',false) inplace of “XXX”

(if that will work might depend on the location you are using it in)

there might also be the possibility of adding `global $post;’
and then use $post->post_title inplace of “XXX”

have you tried something like:

<?php global $post;
$title = $post->post_title;
echo do_shortcode('[wp-blogroll catname=$title]'); ?>

(untested)

is ‘the_title()’ the title of a page or post?
and where do you try to run this code?
header?
sidebar?

it may need a ‘wp_reset_query();’ at the start of the code, to reset any distortions of the query_string through the main content.
also added a test line to see the content of $title, to check if it got the right title:

<?php wp_reset_query();
global $post;
$title = $post->post_title;
echo $title; //for testing only, remove later
echo do_shortcode('[wp-blogroll catname=$title]'); ?>

(today is national wp-reset-query day ; lol 😉

another test:
add the title of the page manually into the variable, to check if the shortcode call works with a variable:

<?php global $post;
$title = 'Title of the Page';
echo do_shortcode('[wp-blogroll catname=$title]'); ?>

another try (i just realized that the do_shortcode() function is expecting a string, which my earlier suggestion does not deliver)

<?php wp_reset_query();
global $post;
$title = $post->post_title;
echo $title; //for testing only, remove later
echo do_shortcode('[wp-blogroll catname="' . $title . '"]'); ?>

if that fails:
is the short code self-made or from a plugin?
if plugin, can you post the download link of the plugin?
if self-made, could you paste the code into a http://wordpress.pastebin.com/ and post the link to it here?