The challenge in answering this question is that we do not have most of Origen's original work. Proponents of reincarnation argue that translators and scribes made Origen's works more palatable to orthodox Christianity, and excised the most significant pro-reincarnation sections of his works.
One quote commonly attributed to Origen does not appear to exist in any of his works as we have them today:
The soul has neither beginning nor end. [They] come into this world strengthened by the victories or weakened by the defeats of their previous lives.
However, there are other portions of modern translations of Origen's writings that proponents say are evidence of his belief in the pre-existence of the soul and in reincarnation. Opponents of reincarnation often argue that these are taken out of context and do not actually indicate that Origen believed in reincarnation.
Evidence for his belief in the pre-existence of the soul is seen in the following passage, from Against Celsus, I.xxxii:
Or is it not more in conformity with reason, that every soul, for certain mysterious reasons (I speak now according to the opinion of Pythagoras, and Plato, and Empedocles, whom Celsus frequently names), is introduced into a body, and introduced according to its deserts and former actions?
In the following passage, from De Principiis III.iii.5, he is seen to argue that disembodied souls have free will, and that their natural tendencies to good or evil in this state have bearing on their bodily lives:
To all which instances, those who maintain that everything in the world is under the administration of Divine Providence (as is also our own belief), can, as it appears to me, give no other answer, so as to show that no shadow of injustice rests upon the divine government, than by holding that there were certain causes of prior existence, in consequence of which the souls, before their birth in the body, contracted a certain amount of guilt in their sensitive nature, or in their movements, on account of which they have been judged worthy by Divine Providence of being placed in this condition. For a soul is always in possession of free-will, as well when it is in the body as when it is without it; and freedom of will is always directed either to good or evil. Nor can any rational and sentient being, i.e., a mind or soul, exist without some movement either good or bad. And it is probable that these movements furnish grounds for merit even before they do anything in this world; so that on account of these merits or grounds they are, immediately on their birth, and even before it, so to speak, assorted by Divine Providence for the endurance either of good or evil.
Similar arguments are made in De Principiis III.v.4:
I am, indeed, of opinion that, as the end and consummation of the saints will be in those (ages) which are not seen, and are eternal, we must conclude (as frequently pointed out in the preceding pages), from a contemplation of that very end, that rational creatures had also a similar beginning. And if they had a beginning such as the end for which they hope, they existed undoubtedly from the very beginning in those (ages) which are not seen, and are eternal. And if this is so, then there has been a descent from a higher to a lower condition, on the part not only of those souls who have deserved the change by the variety of their movements, but also on that of those who, in order to serve the whole world, were brought down from those higher and invisible spheres to these lower and visible ones.
These quotes (also available in Ante-Nicene Fathers, Volume IV), however, only support the pre-existence of souls--not necessarily reincarnation.
For reincarnation itself, proponents refer to other sources. They cite Jerome's letter to Avitus, section 7 (also in Select Library of Nicene and Post-Nicene Fathers), in which Jerome quotes the (now lost) Greek text of De Principiis:
And again: "but perhaps this coarse and earthly body ought to be described as mist and darkness; for at the end of this world and when it becomes necessary to pass into another, the like darkness will lead to the like physical birth." In speaking thus he clearly pleads for the transmigration of souls as taught by Pythagoras and Plato.
Similarly, section 15 of the same letter of Jerome, again quoting Origen:
The following passage is a convincing proof that he holds the transmigration of souls and annihilation of bodies. "If it can be shown that an incorporeal and reasonable being has life in itself independently of the body and that it is worse off in the body than out of it; then beyond a doubt bodies are only of secondary importance and arise from time to time to meet the varying conditions of reasonable creatures. Those who require bodies are clothed with them, and contrariwise, when fallen souls have lifted themselves up to better things, their bodies are once more annihilated. They are thus ever vanishing and ever reappearing."