Hasan and Hussain are heavenly names
On the 7th day of his birth, the Holy Prophet (Sallallahu alaihi wa sallam) named him as Hussain.
It is narrated on the authority of Hadhrat ‘Ali (May Allah be well pleased with him) that he had named his elder son as Hamzah and had named Imam Hussain (May Allah be well pleased with him) after his uncle Ja’afar. Then the Holy Prophet (Sallallahu alaihi wa sallam) named them as Hasan and Hussain. (Mo’jam Kabeer, Hadith No: 2713)
Hasan and Hussain are from among the names of paradise. Before Islam, the Arabs had not used these names. As recorded by Imam Ibn Hajr Haitami Makki (May Allah shower His mercy on him) in As Sawaaiq Al Muhriqa, Pg No. 115 and Tarikh Ul Khulafa, Pg No. 149.
When the Imams were born, the Holy Prophet (Sallallahu alaihi wa sallam) said the Adhaan in both their ears. As given M‘ojam Tabarani, Hadith No: 921, 2515:
The Holy Prophet (Sallallahu alaihi wa sallam) performed the ‘Aqeeqa for both of them.
Translation: It is narrated on the authority of Hadhrat ‘Abdullah bin ‘Abbas (May Allah be well pleased with them) that the Holy Prophet (Sallallahu alaihi wa sallam) slaughtered one goat each in the ‘Aqeeqa of the Imams Hasan and Hussain. (Sunan Abu Dawood, Pg No. 392; Sunan Baihaqui, Hadith No: 1900; Tabarani, Hadith No: 1172, 2504, 2503)