In the Shiva Purana 6.9.14 we have:
रुर्दुःखं दुःखहेतुं वा तद् द्रावयति यः प्रभुः ।
रुद्र इत्युच्यते तस्माच्छिवः परमकारणम् ॥
‘Ruḥ’ means misery or the cause of misery. This is destroyed, melted away, by Rudra. Hence the Supreme Lord, Shiva, who is the Ultimate Cause is called ‘Rudra’.
In the Shatapatha Brahmana of YajurVeda we have:
यद्रोदोत् तस्माद्रुद्र।
He cries therfore he is Rudra |
In the Shatapatha Brahmana in the Brihadaranyaka(3.9.4) section we have:
कतमे रुद्रा इतिः दशेमे पुरुषे प्राणाः आत्मैकादशः ते यदास्माच्छरीरान्मर्त्यादुत्क्रामन्त्यथ रोदयन्ति, तद्यद्रोदयन्ति तस्माद्रुद्रा इति ।।
‘Which are the Rudras ?’ ‘The ten pranas in the human body, with the Atman as the eleventh. When they depart from this mortal body, they make one’s relatives weep. Because they then make them weep, therefore they are called Rudras.’
In the Atharvasiras Upanishad we have:
अथ कस्मादुच्यते रुद्रः यस्मादृषिभिर्नान्यैर्भक्तैर्द्रुतमस्य
रूपमुपलभ्यते तस्मादुच्यते रुद्रः ।
He is “Rudra” because his fast moving form is not understood by all but only by great sages and devotees.
In the Katha Aranyaka we have:
यद् रुवन्न् अभ्यवदत् तद् रुद्रस्य रुद्रत्वम्
Because roaring he yelled at them that is Rudra’s fierceness (rudratvam)
Adi Shankara while commenting on the name 'Rudra' in Vishnu Sahasranama cites the above Shiva Purana verse and also gives further more etymologies as well which are as:
Rudra because he is destroyer:
संहारकाले प्रजा: संहरन् रोदयतीति रुद्र:।
He makes people weep by destroying during Samhara Kala (destruction time), thus he is called Rudra।
Rudra because he gives voice (rud):
रुदं राति ददातीति वा रुद्र:।
He gives voice (ie Rud) so he is called Rudra।
Rudra because he removes suffering and cause of suffering:
रुर्दु:खं दुःखकारणं वा, द्रावयतीति वा रुद्र:।
Ru is suffering or cause of suffering and he who destroys it is Rudra
Rudra because he removes cry (Rodana) of Samsara:
रोदनाद् द्रावणाद्वापि रुद्र इत्युच्यते।
He drives away the Rodana (cry) thus called Rudra
Lord Krishna in Harivamsha Parva of Mahabharata gives a beautiful verse which encapsulates many of these etymologies as:
रुद्रो देव त्वं रुदनाद् रावणाञ्च
रोरुयमाणो द्रावणाच्चवातिदेवः ।
Lord Rudra, You are the one who cry, who makes other cry and who removes the cry of Samsara.