Elephants, mammoths and mastodons in the Americas are generally believed to have gone extinct before Book of Mormon times, making the Book of Mormon reference to elephants in early Jaredite times possibly on the late side. While pockets of mastodons may have survived longer than generally thought, we may do well to consider another member of the elephant family, the Gomphotheres, which may have persisted in the Americas during Book of Mormon times. Here is an excerpt from Wikipedia's article on Gomphotheres:
The Gomphotheres are a diverse group of extinct elephant-like animals (proboscideans) that were widespread in North America during the Miocene and Pliocene epochs, 12-1.6 million years ago. Some also lived in parts of Eurasia and Beringia, and following the Great American Interchange, in South America. From about 5 million years ago onwards, they were slowly replaced by modern elephants, but the last South American species did not finally become extinct until as recently as 400 CE.The cited references above are (1) Palmer, D., ed. (1999). The Marshall Illustrated Encyclopedia of Dinosaurs and Prehistoric Animals. London: Marshall Editions. pp.239–242. ISBN 1-84028-152-9, and (2) Prado, J. L.; Alberdi, M. T.; Azanza, B.; Sánchez, B.; Frassinetti, D. (2001), "The Pleistocene Gomphotheres (Proboscidea) from South America: diversity, habitats and feeding ecology", in Cavarretta, G.; Gioia, P.; Mussi, M. et al., The World of Elephants - Proceedings of the 1st International Congress, Rome October 16-20 2001, Rome: Consiglio Nazionale delle Ricerche, pp. 337–340, ISBN 88-8080-025-6, http://www.cq.rm.cnr.it/elephants2001/pdf/337_340.pdf, retrieved on 25 July 2008.
Gomphothere remains are common at South American Paleo-indian sites. One example is the early human settlement at Monte Verde, Chile, dating to approximately 14,000 years ago.
A related reference of possible interest might be the Hueyatlaco section of "The Solutrean Connection Question" by H. Blaine Ensor.
For pictographic evidence of elephants in the Americas, see "The Elephant and the Book of Mormon" by Glen Chapman.
A hat tip to A.L., who kindly send me an article about gomphotheres eaten by humans at Monte Verde.
Whew, I'm glad I got through a whole blog post without mentioning Prop. 8!