I did some testing and with English as the language O'Brien is seen as a unit and "Brien" alone does not appear when I look in sys.dm_fts_index_keywords_by_document.
If I set the language to Swedish or 0x0, I get hits when searching on Brien alone. But I can see that would have other repercussions for you.
I assume that you have control over what is actually passed to SQL Server. That is, you could inject something to the search term? An idea is that you regularly run
SELECT substring(display_term, 3, len(display_term)
FROM sys.dm_fts_index_keywords_by_document(db_id(),
object_id('YourTable'))
WHERE display_term LIKE 'o''%'
You save that list in your application, and when the user enters a search condition, you parse the string and an OR condition on behalf of the user.