Question
Who is Rudra deva?
Answer
The reason why Shiva is called Rudra is described in this chapter of the Shatapatha Brahmana of the Rig Veda:
[Brahma] said to him, 'My boy, why criest thou, when thou art born out of labour and trouble?' He said, 'Nay, but I am not freed from (guarded against) evil; I have no name given me: give me a name!' Hence one should give a name to the boy that is born, for thereby one frees him from evil;--even a second, even a third (name), for thereby one frees him from evil time after time.
He said to him, 'Thou art Rudra' And because he gave him that name, Agni became suchlike (or, that form), for Rudra is Agni: because he cried (rud) therefore he is Rudra. He said, 'Surely, I am mightier than that: give me yet a name!'
He said to him, 'Thou art Sarva.' And because he gave the him that name, the waters became suchlike, for Sarva is the waters, inasmuch as from the water everything (sarva) here is produced. He said, 'Surely, I am mightier than that: give me yet a name!'
He said to him, 'Thou art Pasupati.' And because he gave him that name, the plants became suchlike, for Pasupati is the plants: hence when cattle (pasu) get plants, then they play the master 3 (patîy). He said, 'Surely, I am mightier than that: give me yet a name!'
He said to him, 'Thou art Ugra.' And because he gave him that name, Vâyu (the wind) became suchlike, for Ugra is Vâyu: hence when it blows strongly, they say 'Ugra is blowing.' He said, 'Surely, I am mightier than that: give me yet a name!'
He said to him, 'Thou art Asani.' And because he gave him that name, the lightning became suchlike, for Asani is the lightning: hence they say of him whom the lightning strikes, 'Asani has smitten him.' He said, 'Surely, I am mightier than that: give me yet a name!'
He said to him, 'Thou art Bhava.' And because he gave him that name, Parganya (the rain-god) became suchlike; for Bhava is Parganya, since everything here comes (bhavati) from the rain-cloud. He said, 'Surely, I am mightier than that: give me yet a name!'.
He said to him, 'Thou art Mahân Devah (the Great God).' And because he gave him that name, the moon became suchlike, for the moon is Pragâpati, and Pragâpati is the Great God. He said, 'Surely, I am mightier than that: give me yet a name!'
He said to him, 'Thou art Îsâna (the Ruler).' And because he gave him that name, the Sun became suchlike, for Îsâna is the Sun, since the Sun rules over this All. He said, 'So great indeed I am: give me no other name after that!'
This chapter of the Srimad Bhagavatam gives the same explanation:
There was much anger generated in the mind of Brahmā, which he tried to control and not express. Although he tried to curb his anger, it came out from between his eyebrows, and a child mixed blue and red was immediately generated. After his birth he began to cry: O destiny maker, teacher of the universe, kindly designate my name and place. The all-powerful Brahmā, who was born from the lotus flower, pacified the boy with gentle words, accepting his request, and said: Do not cry. I shall certainly do as you desire. Thereafter Brahmā said: O chief of the demigods, you shall be called by the name Rudra by all people because you have so anxiously cried. My dear boy, I have already selected the following places for your residence: the heart, the senses, the air of life, the sky, the air, the fire, the water, the earth, the sun, the moon and austerity.... My dear boy Rudra, you have eleven other names: Manyu, Manu, Mahinasa, Mahān, Śiva, Ṛtadhvaja, Ugraretā, Bhava, Kāla, Vāmadeva and Dhṛtavrata.
So the original meaning of Rudra is "the crier", because Shiva cried as soon as he was born. Of course, just like the names Vishnu and Narayana (which I discuss here), various mystical interpretations of the name have arisen that have little connection to its etymological origins.
Note: “The question: Who is Rudra deva?” is licensed by Stack Exchange Inc (https://hinduism.stackexchange.com/); user contributions licensed under CC BY-SA.