Question:

Will the resurrection of the dead be put into effect by Elijah the Prophet or by Moshiach?

Answer: 

There are sources who attribute the resurrection of the dead to Elijah and some to Moshiach. In Pirkei D'Rabbi Eliezer it is written that Moshiach will resurrect the dead: "Why is his name called Yinon? Since in the future, he will yinon [cause to flourish and rejuvenate] those who dwell in the dust." (The word yinon is translated according to the interpretation of Rabbi David Luria.) Rabbi Menachem bar Shlomo, the Meiri, likewise writes, "King Moshiach will resurrect the dead."

However, at the conclusion of Tractate Sotah it is written, "The resurrection of the dead will be through Elijah the Prophet of blessed memory, may he speedily come and redeem us." The Tosephot in the beginning of Tractate Taanit writes that the key for the resurrection of the dead was given to Elijah (and his student Elisha).

Based on these various sources it appears that both Moshiach and Elijah will play a role in resurrecting the dead.

As for the reason that Elijah will be involved in resurrecting the dead, Rabbi Joseph Engel in his book "Glosses on the Talmud" explains: First, he had already succeeded in resurrecting the dead. His student Elisha also received these powers from him, and likewise resurrected the dead. Second, he himself "overpowered" death and rose to heaven with his physical body.

Regarding the second explanation, the Lubavitcher Rebbe comments: The phenomenon of death arises from the coarseness of the body. The resurrection of the dead will come about through a refinement of the physical body, so that it will be able to contain the G-dly energy and be eternal. Since Elijah already succeeded in refining his body to that extent, he will extend a similar benefit to all physical bodies.