ANSWER: The Bible itself is silent regarding how Mary and Lazarus died. The last mention of Mary in Scripture is her meeting regularly with the disciples in Jerusalem, after the ascension of Christ, for prayer (Acts 1:14). What we know 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 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 died. There is a possibly problem, however, with the idea that Mary lived out her life in Jerusalem and died in the city.
The Bible is clear that the apostle John was charged directly by Christ on the cross to take care of his mother (something which, as a side note, would not be needed if Mary's husband Joseph was still living - John 19:26 - 27). How long John and Mary stayed in Jerusalem after Christ's death is not known. We do know, however, that in all likelihood (based on Foxe's Book of Martyrs) John was the person who started churches in Smyrna, Pergamos, Sardis, Philadelphia, Laodicea, and Thyatira (six out of the seven churches of Revelation). All these churches were located in Asia Minor and were fairly close to Ephesus. John's long-time residence in the city is well-known.
In fact, 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 more plausible that John 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 he had died, 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).
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.