Mentioned by Adi Shankaracharya in his commentary of Brihadaranyaka Upanishad 1.3.1
- स्वेषु एव असुषु रमणात् सुरेभ्य:
The one who indulge in one’s own असु (vital senses) is called Asura
- यस्मात् दृष्टप्रयोजनज्ञानकर्मभाविता असुरा:
The one who is guided by what is seen and what is concluded, is called as Asura
गता: असव: येषां ते – गतासव: ( बहुव्रीहि )
Another reference from Ramayana
In Ramayana Balakanda chapter 45 , When Rishi Vishwamitra (विश्वामित्र) was telling Lord Rama the details of Samudra Manthan (समुद्रमन्थन), he said that during the churning of ocean the Varuni (वारुणी) was born out of Samudra but Asuras did not accepted her, whereas Devata accepted her.
Therefore Asuras got the name, as they are without Sura (असुरा)
दिते: पुत्रा न तां राम जगृहुर्वरुणात्मजाम्।
अदितेस्तु सुता वीर जगृहुस्तामनिन्दिताम्।।1.45.36।।
O Rama ! while Diti’s sons did not accept the Varuni, but the sons of Aditi did received her.
हृष्टा: प्रमुदिताश्चासन् वारुणीग्रहणात्सुरा:।।1.45.37।।
For that reason, sons of Diti were called ‘Asuras’. Aditi’s sons were known as ‘Suras’. Devatas grew exceedingly glad for having Varuni.