According to most Christian traditions, God cannot change His mind.
There are those that teach that He can change His mind, a position inherent in Open Theism. However, this is seen as a heresy by most orthodox Christian traditions. It denies the omniscience of God.
In recent years though a new movement has emerged called Open Theism.
This movement has taken Arminianism to its logical intellectual
conclusion. Knowing the "problem" that complete knowledge of the
future means that the future is fixed, they have consciously taken the
position that God does NOT know the future at all. They argue that
because the future does not yet exist, even God does not know of it.
In layman's terms, if He were to change His mind, it would mean that He was mistaken in the first place. This flies in the face of the doctrines of Omniscience and infallibility.
There are several passages in Scripture where God appears to change His mind, for example, in cases where He has promised judgment and then "repents". In those cases, the people repent and turn toward God, and God spares them the judgment He had promised.
I these cases, He is not changing his mind. had they continued in their sins he would have carried out the promised judgment. However, since the people repented, He was able to show his mercy. Further, since God knows the beginning from the end, He knew in advance that they would repent. He knew that his judgment would not be necessary.
There's a fine article explaining this in further depth at Grace Online Library.
The biblical narratives in which God appears to repent, or change His
mind, are almost always narratives that deal with His threats of
judgment and punishment. These threats are then followed by the
repentance of the people or by the intercessory petitions of their
leaders. God is not talked into “changing His mind.” Out of His
gracious heart He only does what He has promised to do all along – not
punish sinners who repent and turn from their evil ways. He chooses
not to do what He has every right to do.
This is consistent with the Catholic teaching as well.
In hearing our prayer God does not change His will or action in our
regard, but simply puts into effect what He had eternally decreed in
view of our prayer. This He may do directly without the intervention
of any secondary cause as when He imparts to us some supernatural
gift, such as actual grace, or indirectly, when He bestows some
natural gift. In this latter case He directs by His Providence the
natural causes which contribute to the effect desired, whether they be
moral or free agents, such as men; or some moral and others not, but
physical and not free; or, again, when none of them is free. Finally,
by miraculous intervention, and without employing any of these causes,
He can produce the effect prayed for.
And Reformed Theology:
In reading certain narrative portions of Scripture, some have
incorrectly concluded that God changes His mind. Yet the Bible is
clear that not only does God not change in His essential nature (Mal.
3:6) but that He does not repent or change His mind. The Bible
actually teaches this in a didactic portion. "God is not man, that he
should lie, or a son of man, that he should change his mind. Has he
said, and will he not do it? Or has he spoken, and will he not fulfill
it?" Numbers 23:19.
For the sake of argument though, lets try to imagine God literally
changing His mind. I want to explain how this concept is inseparably
linked with God's omniscience because for God to change His mind, He
would need to make a decision and then be given new information He did
not have before, so that He could either see the error of His ways, or
choose a better course of action. It is important we see this.For God
to change His mind, it would mean that God is learning new material as
each day unfolds, and because you and I make that information known to
God, or He sees that plan A is not working too well, because He is now
armed with new information, He can make a better decision than the one
He did previously. However, this idea would totally undermine God's
exhaustive knowledge of future events (His omniscience) one of the
very attributes of God. Such a thought is unthinkable. He would not be
the all knowing God Scripture declares Him to be if indeed He ever
learnt something. No, He has always had total, complete and infinite
knowledge of all things from all eternity past.
Thus, the answer would appear to be "no" from the perspective of the major branches of Christianity.