Paramshiva is called as Mrityunjaya because he has conquered the death. As per the tales of Puranas, Shiva is called Mrityunjaya due to his victory over the Yamraj (God of death). Bhatt Bhaskar precisely explains the meaning of Mrityunjaya—
Purport: O Absolute one! Millions of Brahma, Vishnu, Rudra and other demigods are engulfed by death, but you are not subjected of death, so you are known as Mrityunjaya.
The divinity of the word ‘Mrityunjaya’ is highly contemplative for the seekers. In the scriptures ignorance is described as Mrityu (death). Shiva is never influenced by ignorance. So he is Mrityunjaya.
In Sanat-Sujateeya, Pramad (carelessness) is also taken as Mrityu. Shiva is free from such indolence and also instructs his devotees to stay alert; therefore he is called as Mrityunjaya. One who may help his devotees to cross the ocean of death, certainly he must be Mrityunjaya.
In the opinion of Vedanta, Mrityu is said to be the name, form and action. Lord Shiva who is the prime support, he is never influenced by these dependents; rather he brings consciousness in them through his knowledge. Therefore he is named as Mrityunjaya.
According to Tantra, life-air is supposed to be the Rudra. It is impossible to stay aloof from life-air. Therefore Rudra is called as Mrityunjaya. As per minute perception, dualism (Dvaita) is the cause of death. One can gain victory over death through the concept of oneness (Advaita). Shiva is always fixed in the sense of oneness, therefore he is Mrityunjaya. Death is certain for the follower of dualism. Only a self-realized person can control the inevitable death. Shiva is never captured by the illusive cycle of time, such assumption is the true sense of worship of Mrityunjaya.