Is it permissible to delay Fajr prayer in the mosque in order to wake up a younger brother, knowing that he cannot go to the mosque by himself?

It is permissible to wait for one's brother to wake him up so he can accompany him to the mosque for Fajr prayer, even if this occasionally results in his being late and missing a rak'ah. For, catching one rak'ah with the Imam is considered to be catching the congregation. It is, however, recommended to try and wake the brother early to catch the opening takbeer and attain the full reward.