Is it considered showing off (riya') to memorize the Qur'an and participate in a competition with financial prizes for the purpose of encouragement, given that the primary goal is to memorize the Qur'an and the money is a secondary means to help the family?

There is no harm in taking money for memorizing the Quran, as long as memorizing the Quran is the primary goal, and money is a means of encouragement to achieve that goal. The ruling depends on the original motive for memorization: whether it is seeking the countenance of Allah or seeking money.