Of course, it's an easy objection to answer: if you ask them, most of the objectors don't use the KJV in their regular Bible reading. It's much more likely to be the NIV or the NRSV, where the phrase Jesus used is translated "do not keep on babbling like pagans" or "do not heap up empty phrases as the Gentiles do" - and patently the Jesus Prayer isn't anything like that.
But for all the ease with which one can refute such proof-texting, our objectors do have a point. Occasionally you will find Christian writers, whether in approval or disapproval, referring to the Jesus Prayer (as well as prayers like the Hail Mary, and perhaps even the Kyrie) as a "mantra", by which they seem to mean a phrase that is repeated over and over again, more or less regardless of meaning, in order to bring about some psychological effect, such as reducing stress or "emptying the mind." And of course the Jesus Prayer is not that. Unlike many of the mantras sometimes used by practitioners of transcendental mediation and similar paths, that are also often given in languages unfamiliar to the user, the Jesus Prayer is a prayer.
Almost all the teachers of the Jesus Prayer whom I have encountered make the point somewhere, though they may have different ways of putting it, that the key to this way of praying is intentionality. We mean what we say, and our using it repetitively is much more like the prayer of Bartimaeus the blind man, who "was sitting by the roadside begging. When he heard that it was Jesus of Nazareth, he began to shout, 'Jesus, Son of David, have mercy on me!' Many rebuked him and told him to be quiet, but he shouted all the more, 'Son of David, have mercy on me!'" (Mark 10:46 NIV)
In its simplicity and its self-abandonment, the Prayer comes to resemble, too, the prayer of the tax collector at the temple, who "stood at a distance. He would not even look up to heaven, but beat his breast and said, 'God, have mercy on me, a sinner.'" (Luke 18:13 NIV) (The prayers of Isaiah 6, and Revelation 4:2 and 5:11-14 are prayers of repetition also, but of praise rather than of supplication or intercession.)
It is best to approach saying the Jesus Prayer with as few preconceptions as possible. Although I have read widely, and I hope deeply, on the Prayer over the years, I began saying it when I knew very little of the tradition, or the traditional methods, of praying the Prayer. It took hold, as God had obviously intended it should, and became simply part of who I am before God. In fact, although when I was first introduced to the Prayer by Fr. Francis Horner SSM back in 1978, he gave me Per-Olof Sjogren's wonderful book to read, a good deal of what happened in the years following were things for which I had no frame of reference. I only discovered much later that they were commonplace in the experience of those who pray the Prayer.
So we don't need to be afraid, if God calls us on this way of knowing him, to strike out into the deep. After all, even the best maps can do no more than hint at destinations, and maybe warn of shoals; they can convey nothing of the sea-wind, the endless cry of the gulls, the wonderful scent of the waves as they break, or the peace there is in the lift and rock of the deep-water swell...