How to add classes to h2 block title tags in sidebar region
template_preprocess_block()

Inspect the block object in this preprocess function. You could use either the 'module' and 'delta' properties to target a specific block or the 'region' property to target all blocks in a certain region.

H2 css-classes are listed in the title_attributes_array. The code below shows how to add a new class. This example is based on a block generated by Views:

function MYTHEME_preprocess_block(&amp;$variables) {
  $block = $variables['block'];
  if ($block-&gt;module == 'views' &amp;&amp; $block-&gt;delta == 'name_of_my_view-block') {
    $variables['title_attributes_array']['class'][] = 'my-new-class';
  }
}


Sidenote: dpm() by the Devel module is a handy tool for printing out the available data in variables:

function MYTHEME_preprocess_block(&amp;$variables) {
  dpm($variables);
}