ANSWER: The Bible itself is silent regarding how did Mary and Lazarus die. The last mention of the mother of Jesus, in Scripture, is her meeting regularly with the disciples in Jerusalem, after the ascension of Christ, for prayer (Acts 1:14). What we believe happened to her is mostly based on tradition.
Roman Catholic tradition, found in its apocryphal works dating back to the second century A.D., holds that after the death of her firstborn son Jesus that Mary (the sister of Lazarus) lived out her life in Jerusalem. The place venerated as her tomb is supposedly located near Jerusalem in the valley of Kidron. Nothing is stated regarding the cause of her death, how old she was or what year she perished. There is a major problem, however, with the idea that Mary lived out her life in Jerusalem only to die in the city.
The Bible clearly records that the apostle John was charged directly by Christ, while he was on the cross, to take care of his mother (something which, as a side note, would not be needed if her husband Joseph was still living, see John 19:26 - 27). How long John and Mary stayed in Jerusalem after Christ's death is not known. What is known is that John spent a great deal of his time at Ephesus, which is in Asia Minor.
It is widely held (even among the Catholics) that John, after writing the book of Revelation and his release from being banished to the island of Patmos, lived out the remainder of his days in Ephesus. Given that the bulk of his pastoral work was in Asia Minor, not Palestine, it seems plausible that he took Mary with him as he labored spreading the gospel and that she spent the last years of her life in Ephesus.
Lazarus was the brother of Mary and Martha. The story of his death, why Jesus seemed unconcerned and uncaring when told about his death, and the miracle of resurrecting him in the face of many who hated him, is a fascinating story of itself (see John 11:1 - 4, 6, 17, 20 - 21, 23 - 27, 32 - 34, 39 - 47). There is no known Biblical reference to the death of Lazarus a second time after being resurrected from the grave. The Chief priests and other religious leaders, however, certainly WANTED to kill him a second time because his resurrection was causing many Jews to leave Judaism and begin to believe in Christ (see John 12:9 - 11, 17 - 19).
Catholic tradition states Lazarus was the first Bishop of Marseilles and that he died somewhere in the second half of the first century A.D. The Greek church claims his body was eventually brought to Constantinople and buried there. No other information exists.