Linda Ellis sent me this picture of a trout lily with eight petals on one flower and six on the other. She did it just to send me "down the rabbit hole" of botanical oddities in search of a cause. I finally emerged with the help of more knowledgeable friends.
When searching flowers with extra petals, most references are to "double-flowered" plants such as roses and carnations. These are flowers inside of flowers, giving the appearance of a complete reproduction of the blossom. Most of these are sterile and are propagated by cuttings as their normal stamens have been re-purposed by the plant into a new set of petals. There is a good video on flower development at USCD.edu that explains the genetics and this link demonstrates how flowers develop.
On the other hand, Linda's trout lily was a single specimen with two extra petals. This can be common in some species where a flower can have 5-10 petals, such as rue anemone, Thalictrum thalictroides, as discussed in a past blog .
Dr. Michelle Bowe* explained that it is likely a meristem mutation in the flower bud itself. The meristem is undifferentiated tissue in areas of the plant where growth can take place. If the mutation was in the whole plant, both flowers would have 8 tepals.
What I called six petals are actually tepals, three petals and three petal-like sepals. Wikipedia explains "A tepal is the term used to identify one of the outer parts of a flower (collectively the perianth) when these parts cannot easily be divided into two kinds, sepals and petals.
If the two stems both had eight tepals, the mutation would be in the stem itself. Mutations are just a normal part of life, and sometimes it may just be one gene (like the gene that controls the flower bud). This is important to remember the next time you think you have identified a flower but it has the "wrong" number of petals, tepals, or whatever.
* Professor of Biology, Missouri State University
A more detailed technical explanation from Dr. John Heywood is here.