Add the title of a widget as an ID – for anchor links

Yes you can add an around the title of the widget, you do this in the register_sidebar code which you would put in your functions.php

add_action( 'widgets_init', 'theme_register_sidebars' );
function theme_register_sidebars() {
   'id' => 'mysidebar-sidebar',
   'name' => __( 'My Sidebar', 'themename' ),
   'description' => __( 'Widgets for my sidebar', 'themename' ),
   'before_widget' => '<div class="widget">',
   'after_widget' => '</div>',
   'before_title' => '<h2 class="widget-title"><a id="about">',
   'after_title' => '</a></h2>'

You create the new “widget area” or sidebar which you then put in actual widget in.

jQuery Version (not tested):

 $( '#id h2.widget-title').wrapInner( "<a id="about"></a>" );

You would then add the jQuery either to a JS file and Enqueue it or put it into wp_footer using the filter of the same name (link)