I Believe that the Mon or coat of arms isn't, well in a sense, unique specifically to its clan. so like for example, how did the Oda family establish it's clan mon to be a flower or how about the Hojo clan's dragon scale mon (which we now know affectionately as the triforce.)
It is believed they started as patterns on the palanquins, ox carts, and clothing of court nobels around the 12th century. The motifs were usually plants and they started out being realistic design which became more abstract. They were not initially family crests, however children tended to choose patterns similar to their parents, and so in that way became associated with the family.
Kamons are very interesting artistically because they tend to create unique design by "variation on a theme" rather than by just inventing a whole new design. A paulownia blossom (kiri) can be made into dozens of unique designs. Including korin-style, herons, butterflies, ships, and the paulownia bat, my fave. http://rlv.zcache.com/bat_shaped_paulownia_keepsake_boxes-rc16db811a485488991c7fc25935b1334_aglbn_8byvr_152.jpg
More examples of the paulownia theme: http://www.harimaya.com/o_kamon1/zukan/mon/kiri100.jpg
Bisected: http://smt.blogs.com/photos/uncategorized/kiri.jpg
