All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Saying 'the prayer is better than sleep' is a Sunnah in the Fajr Athaan. Ibn Qudaamah said: 'It is a Sunnah to say in the Fajr Athaan 'the prayer is better than sleep''". This could be said at the first and the second Athaan of Fajr. An-Nawawi said: 'The scholars of our school [Shaafi'ee] are of the view that it is permissible to say 'the prayer is better than sleep' in every Fajr Athaan whether before the actual time of Fajr or after its beginning time. Al-Baghawi said: 'According to the most correct opinion if one says 'the prayer is better than sleep' in the first Athaan, he should not say it in the second.'
However, the Maaliki and Hanbali schools are of the view that it is permissible to say 'the prayer is better than sleep' in the first and second Fajr Aathaans.
Allaah Knows best.