First of all, it should be noted that Tarawih Prayer is a nafl (a sunnah) that is highly recommended. There is nothing wrong for a minor who knows the rules of salah and has good knowledge of the Qur’an to lead this Prayer.
In his comment on this issue, Sheikh Ahmad Kutty said:
” Prayer is the best and the most meritorious of all deeds in Islam. Prayer has both a form and a spirit. The form of it is the precise manner and method, which involves performing the same with all of the essentials, conditions, and manners; while the spirit of it entails awareness of standing before Allah, and understanding what one is reading and saying in Prayer. While choosing an imam to lead us in Prayer, we should do so while keeping the above in mind. The criteria for choosing an imam are; knowledge of the Qur’an (something inseparable from understanding), knowledge of the Sunnah, and piety.
Coming to the issue of having a minor leading the salah if he is well-versed in the Qur’an and fully conscious of the rules of salah, scholars of various Schools of Jurisprudence differ on this issue:
Generally speaking, scholars of Shafi`i and Hanbali Schools are of the view that Prayer of a minor is valid provided he is knowledgeable and conscious of the rules of salah; while the scholars of the Hanafi and Maliki Schools are of the view that it is not permissible in fard Prayers (obligatory five daily Prayers), while it may be permissible in nafl (optional) Prayers: Tarawih can be included in this category.
As Imam An-Nawawi has pointed out, the first view is supported by a well-attested tradition that states that Prophet Muhammad (peace and blessings be upon him) had appointed someone who was a minor to lead the Prayer because of the minor’s superior knowledge of the Qur’an. This view has been endorsed by a number of scholars from both early and later generations.
In conclusion, there is nothing wrong in putting forward a minor to lead Tarawih Prayer if he is qualified to do so because of his superior knowledge of the Qur’an as long as he is aware of the rules of the salah.