The power of King's metaphor, I would argue, derives (in part, at least), from its connection with biblical imagery. First of all, his reference to a "mountain of despair"--representing the long, seemingly hopeless struggle of African Americans for freedom and dignity--has a rather biblical ring to it. Interestingly, earlier in the speech King quotes Isaiah's prediction that one day "every mountain and hill will be made low" (Isaiah 40:4), which serves to prefigure his later reference to the "mountain of despair." Moreover, in at least one biblical passage, a mountain represents something that is seemingly unmovable, and yet can be moved--when Jesus tells His followers in Matthew 7:20 that if one has sufficient faith, even a mountain can be moved. In the same way, King seems to be saying, the apparently unchanging reality of segregation and its attendant evils can be changed through hope.
Second, the phrase "a stone of hope" resonates with several biblical images. In the Bible, a rock or stone sometimes represents a place where God's people can find refuge, as when David speaks of seeking the safety of the "rock that is higher than I" (Psalm 61:2). In addition, a rock is sometimes associated with God's presence, as when Jacob slept on a pillow of stone and dreamt of a ladder leading up to Heaven, leading him to call the place Bethel, "the house of God" (Genesis 28). Perhaps most striking is King's reference to "[hewing] ...a stone of hope" from a mountain. This recalls for me the prophet Daniel's dream in which a stone is cut out "not by human hands" and then crushes a statue representing all earthly powers (Daniel 2)--long understood by Christians as a prediction of Christ's Second Coming, when He shall judge the nations and rule in power.
That Martin Luther King should employ a metaphor with such biblical resonance is not surprising, given that he was a Baptist preacher by profession. He was, by virtue of his pastoral work, a man well-acquainted with the stirring language of Scripture and was quite skillful in appropriating it for his purposes. In creating such a memorable metaphor with a clear biblical undertone, he helped demonstrate the great and enduring power of the word of God.
Picture of the Martin Luther King, Jr. Memorial in Washington, DC from jesussavesministries.blogspot.com