ANSWER: His name means "God the Lord, the strong Lord" and Elisha means "salvation of God." If you are questioning a blood relationship, there is none. Except for a mention in 1Kings 17:1, we know nothing of his parentage or his early life: "And Elijah the Tishbite, of the inhabitants of Gilead . . . " (1Kings 17:1)
Elisha was a native of Abel-Meholah in the northern kingdom of Israel. His home, however, was fairly near that of Elijah's.
According to Richard J Clifford, Weston School of Theology, Elijah prophesied during the reigns of Israelite kings Ahab (874 - 853 B.C.), Ahaziah (853 - 852 B.C.) and Joram (852 - 841 B.C.). Elisha took over during Jehoram's reign, and prophesied through the reigns of Jehu (841 - 814 B.C.), Jehoahaz (814 - 798 B.C.) and Jehoash (798 - 782 B.C.). Dr. William Smith wrote of him: " . . . he 'stood before' Ahab with the suddenness of motion to this day characteristic of the Bedouins from his native hills . . . "
It seems he disappeared just as quickly. Elijah was a man of few words. His style was to pronounce his message then quickly leave (2Kings 1:4). The kings knew and feared him. When King Ahaziah asked his messengers who had given the message that he would die, they said it was a man who wore a leather belt (2Kings 1:8). The king's troubled response was "It is Elijah the Tishbite." (verse 8)
Elijah led what we might call today a "dynamic life." He was a prophet who warned and punished the Kings of Israel for their sins. He raised a widow's son from the dead (1Kings 17:2-24). He also had a major confrontation with Israel's false prophets on Mount Carmel, which ended in the execution of 850 prophets of Baal and Asherah (1Kings 18:18-40). God even gave him the power to OUTRUN the chariot of Ahab (who also had a head start) to the entrance of Jezreel (1Kings 18:46)!
Elijah's service to God was such that he represented all the Old Testament prophets when Jesus was transfigured and in a vision is seen talking to him and Moses (Luke 9:28 - 36). John the Baptist, who prepared the way for Jesus before his ministry began, preached in his spirit and power (John 1:19 - 23, Matthew 17:10 - 13).
Elisha, on the other hand, was a humble farmer (1Kings 19:16 - 19). However, he apparently knew what it meant when cast over him was his mentor's mantle, as he seems not to have questioned, except to say goodbye to his parents (verses 20-21). He then followed him until the day God took Elijah away by a chariot of fire to another location.
Dr. Smith wrote of Elisha: "In almost every respect [he] presents the most complete contrast to Elijah."
Elijah's tactic was to hit and run. He was rough shod and hairy and thundered his denunciation of kings. Elisha, however, stayed as a visitor in the homes where he went. He dressed like the people of his day and generally succored the people. After sixty years of service as a prophet of the eternal God he died in his own home (2Kings 13:14, 20). God's power, however, showed through him even after his death. For example, some men were burying a man when a gang of men appeared. They threw the body into Elisha's tomb, and when it touched his bones, the man came back to life (2Kings 13:20 - 21)!