An essay, "Eggs alone," in the March 10, 2005 issue of Nature states that normal human females begin their reproductive period with about a million eggs. Ann A. Kiessling, the author, who is at the Harvard Institutes of Medicine, says that the vast majority of these eggs, about 20,000 per year, "die in the ovary." (Volume 434:145--Nature is not freely available on the Internet, but the essay can be obtained here for a fee, and should be available in all university libraries. Nature is among the most important scientific periodicals in English. The original Watson and Crick paper was published there, for instance.)
Some ovarian growths may come from eggs which develop without fertilization. (Such growths occur naturally.) Teratomas, the author says, have some differentiated tissues, indicating that, if they indeed come from unfertilized eggs, such eggs, or the tissue developing from them, might work as if they were human embryonic stem cells. (This article, from eMedicine, indicates that parthenogenesis, or the development of an unfertilized egg, is the most likely cause of such growths, which agrees with Kiessling.) There would be no destruction of human embryos involved in using these as stem cells. As she writes, we would, however, be exchanging "the moral dilemma of using human embryos for obtaining stem cells with the moral dilemma of collecting eggs from the ovaries of women for therapeutic instead of reproductive purposes."
She further writes that, as far as she can see, there would be no moral problems with using such eggs for therapy on the woman who is carrying them, and suggests Type 1 diabetes or spinal-cord injuries as two such applications for therapy from self-derived cells. As she points out, few persons object to storing your own blood for later use in a surgical procedure, and she sees use of egg cells derived from your own body as morally equivalent. It remains to be seen whether this sort of treatment is possible, and would be effective. I would be amazed if there were no moral objections to this, or any other procedure involving human eggs.