Nowadays, many Japanese people, or people from Japanese descent, have lost track of their family insignia. When they try to find their kamon, they generally have only one clue left: their family name.

However, Japan never had strict and enforced rules regarding the use of kamon. Even during the pre-modern times, warriors from different families could be bearing the same mon, as John Dower relates in his The Elements of Japanese Design:

In 1771 a hapless envoy from Choshu was stripped of his garments upon entering Tosa because the crest he wore—his legitimate family crest—was identical with one of great prestige in that latter domain. Several years later this envoy’s son paid a visit to Tosa, and on this occasion it was duly noted that he wore his family’s substitute crest. 

“Substitute crest” refers to what is known as kaemon, fukumon, or uramon, i.e. secondary kamon used in different informal situations. The primary insignia of a lord or a warrior was called jōmon, honmon or omote mon, and wast thus the official one, used in formal situations, when traveling, etc.

Warriors and aristocrates were thus using several different mon at the same time. Those insignias were not necessarily representing the same emblems. The jōmon could be a paulownia mon while the auxiliary mon could well represent a wave, an instect, a tool, etc.

Until the end of the 19th century, kamon (and family names) were officially reserved to the samurai class, the aristocracy and religious institutions, so probably not even 10% of Japan’s population. In fact, from the Edo period, commoners started to use insignias as well. This was against shogunate’s rules, but it was somehow tolerated. In 1870, the newly formed Meiji government promulgated the Heimin Myōji Kyoka Rei, a law officially authorizing every citizen of Japan to bear a family name. Subsequently, the government signed in 1875 the Heimin Myōji hisshō Gimu Rei, a law forcing every citizen to choose a family name: the 1870 edict had not been followed by commoners since registering a family name meant the possibility to be drafted in the new Imperial Army… However, with family names came also the official authorization for everybody to choose and use a kamon.

People then either adopted the mon that their family had used informally so far, or they chose an emblem related to their profession (tools, boats, objects…), some decided to use the jōmon or one of the kaemon of the former lord of the area, etc. Some just designed a new kamon for themselves. This could have meant a new boom in Japanese heraldry, but interestingly, it seems that people mostly pick up common insignias. That led numerous unrelated families to share the same mon. Some kamon were more popular than others, and nowadays, the Jū Dai Kamon, the ten most used emblems are: the Fuji (wisteria), Katabami (creeping woodsorrel), Mokkō (Japanese quince), Tsuta (Japanese ivy), Taka-no-ha (hawk feather), Kashiwa (daimyō oak), Kiri (paulownia), Myōga (Japanese ginger), Omodaka (arrowhead) and Tachibana.

It is therefore almost impossible to determine someone’s kamon just from his family name. One family name can be associated to numerous kamon, and conversely, a unique mon can be shared by numerous families. To give a striking example of this, here is a page from Takasawa Hitoshi’s Myōji Kara Hiku Kamon no Jiten, which lists up all the kamon that can be associated with the name “Harada” (原田):
harada kamon
And this is probably not exhaustive…
So how can someone know his family emblem? The surest way to find out is to look for the family chōchin, a lamp consisting of a candle in a frame of split bamboo covered by paper. The chōchin usually features the family’s name and mon, painted on the paper. The lamp is mainly used during the Bon festival where it guides the spirits of the ancestors back to the household altars. 
choshin with kamon
One’s kamon is also generally carved onto the family grave stone. This is actually the surest way to know which mon is yours.
grave with kamon
If one does not have access to those 2 clues, then it is going to be difficult to know what his family insignia is. There is a third clue, but it is quite unreliable: formal kimono adorned with kamon, called mon-tsuki or “kimono with emblem”. They are mainly worn during weddings or funerals. However, many people in Japan actually buy/rent generic mon-tsuki with one of the Jū Dai Kamon embroidered on it. Therefore, using one’s ancestor’s mon-tsuki to determine a kamon can be completely misleading.