Where is Pathauto's "Strings to Remove" stored in MySQL? I've tried looking in the "variable" table with SELECT name FROM variable but I think it's not there. 

Where does the module save that setting?
It's stored in the variable named pathauto_ignore_words, you can get it directly like this:

SELECT value FROM variable WHERE name = 'pathauto_ignore_words'


or using the Drupal API like this:

$ignore_words = variable_get('pathauto_ignore_words', '');