ILIKE is only case-insensitive, and won't match accented characters. The only thing I can think of doing is to create a function which will replace characters with their equivalent non-accented counterparts and use that in the query.
Example: SELECT name FROM people WHERE unaccent_string(name) ~* 'Pédro';
The function would perform a reg-ex replace similar to: s/[èééêë]/e