I don't think this quite fits under Clan Traditions, but I thought it would be of use to some of the people new to the setting as well.
Also, a lot of it's quite cool.
Great Clans
These 7 clans were formed by the Kami who fell to earth and didn't become the Emperor. Each of them has a family descended from their original Kami, and others from their earlier or later followers
Crab
Hida is the Clan Kami and leading family. There are some offshoots of the family - the Mantis, Badger and Boar minor clans all trace their founders back to a Hida.
Hiruma, Kaui and Kuni worked together to defeat an Oni and were awarded with families because of that. They have no particular offshoots.
The Yasuki joined the Crab after leaving the Crane, the only family to have moved clans (currently). A cast out Yasuki won the right to form the Hare clan.
Crane
Doji is the Clan Kami and leading family. As well as the Daidoji being an offshoot, the Suzume family of the Sparrow descend from them.
Kakita sought and won the hand of Doji in marriage, and was permitted to recruit for a family. One of their children founded the Daidoji.
The Daidoji, as said, are descended from both Kakita and Doji.
Yasuki was a wise woman who helped Kakita win Doji's hand, gaining a family in return.
Asahina was an aggesive Isawa who attacked the Crane. After being shown up by the Crane Champion's daughter, he swore to pacifism and left his original family, and was allowed to found a new one in the Crane, for bringing the secrets of Shugenja
Dragon
Togashi is the Clan Kami and leading family. They have no offshoots.
Agasha and Mirumoto followed Togashi while he sought to gain wisdom, and were awarded families. Both have offshoots. From the Agasha come the Kasuga of the Tortoise and the Kitsuki of the Dragon. From the Mirumoto come the Tonbo of the Dragonfly.
Kitsuki was an Agasha with an exceptionally perceptive mind, who won himself a family by saving the daimyo of the Mirumoto.
Lion
Akodo is the Clan Kami and leading family. The Wasp clan was founded by a member of this family and a Scorpion.
Ikoma boasted of his life to Akodo so well that the Kami took him as an advisor and gave him a family.
Akodo sought out Matsu, a famed warrior, to be his wife, but she refused, even though he won a literal duel with her as she would be No Man's Wife. The son of a Matsu and a Hida founded the Mantis.
The Kitsu used to be a race of leonine sorcerors, who Akodo hunted nearly to extinction as monsters. When it was realised that they were actually intelligent, the last handful were transformed into humans and given a family of their own.
Phoenix
Shiba is the Clan Kami, and technically the leading family. They have no offshoots.
Isawa was the leader of a large settlement of Sorcerors. He led his family to join Shiba after the Kami swore to protect them. Many families trace their heritage to Isawa. The Asahina of the Crane was founded by an Isawa. The Chuda of the Snake and the Moshi of the Centipede. The founder of the Dragonfly was married to an Isawa.
Asako was a great healer, and was so respected by the Isawa and Shiba that she was given her own family. Asako's husband founded the Yogo of the Scorpion Clan.
Scorpion
Bayushi is the Clan Kami and leading family. A Bayushi and an Akodo founded the Wasp clan.
Shosuro was an enigma that Bayushi adopted and gave her own family. OOC we know that Soshi was Shosuro disguised after the first Day of Thunder.
Soshi was a young shugenja who was given a secret purpose and a family by Bayushi. There are no offshoots.
Asako Yogo was cursed by Fu Leng to betray the one he loved the most. He was going to kill himself but Bayushi persuaded him to leave his family and friends to offer his services to the Scorpion Clan, who he had no feelings for.
Unicorn
Shinjo is the clan Kami and leading family.
Ide and Utaku were a married couple who decided to follow and were given families.
Iuchi was a shugenja who joined Shinjo after the war against Fu Leng
Members of all 4 of these families were part of the founding of the Fox Clan.
The Moto family were a group of nomads who joined the Ki-Rin clan during its wanderings outside Rokugan.
Imperial Families
Hantei is the Kami and the only people to take the name are the Emperor and his family. As soon as a new Emperor is acknowledged, all of his siblings join one of the other Imperial families and give up the name.
Seppun was a priestess who witnessed the Kami fall to the earth, and swore fealty to Hantei with all of her people. For a while Otomo was an adopted son.
Otomo was a leader of a tribe that had spent some time fostered with Seppun's. He swore his people's fealty to Hantei shortly after the Seppun family was created. One of his followers founded the Miya family.
Miya was a follower of Otomo. Shortly after the death of Hantei, Otomo manipulated things to have Miya granted the right to found a family.
The Nasu were founded as a whole family of Imperial Bureaucrats during the days of the Gozoku from Crane, Phoenix and Scorpion samurai, but was disbanded when the conspiracy was brought down.
Minor Clans
Badger
Hida Domogu won a trial of strength set by the 2nd Emperor to found a new clan to guard the northern pass out of Rokugan. Hantei Genji gave them the family name Ichiro.
Boar
An expedition of Crab went prospecting into the mountains in the Crab lands, but contact with them was lost. Years later, a Samurai turned up at Imperial Court with back taxes of mined Iron. The Emperor of the time granted them Minor Clan status and the name of their leader Heichi as a family.
Centipede
The family of a sensei of a small Isawa shugenja school became cut off from their Clan. The Otomo convinced the Emperor that they would be better treated as a Minor Clan for administrative purposes and the Moshi family was born.
Dragonfly
A Phoenix Shugenja was promised to a Lion Bushi, but instead she ran away and married a Dragon Bushi instead and was given land by the Dragon. The Lion moved to crush the oathbreakers, but the Dragon and Phoenix Clans combined to prevent it. The Emperor apparently approved, because he named them a minor Clan with the name Tonbo.
Falcon
The ashigaru Hayabusa saved the life of an Imperial Advisor from assassins, and his sons were given control of an unclaimed province and the right to be a Minor Clan, but no family name.
Fox
When the Ki-Rin left the Empire, not all of the members wished to leave. Those who stayed behind were granted the status of Minor Clan and control of the former Ki-rin provinces by the Emperor. They did not have their own family name.
Hare
The ronin Reichin was instrumental in warning of the attack of the first bloodspeaker and his undead horde, and was rewarded with the name Usagi.
Mantis
Kaimetsu-Uo, the son of Hida Osano-Wo and Matsu Kyoda was overlooked as Daimyo in favour of his half-peasant half-brother, so left the mainland to rule in the nearby Islands. This is the origin of the Minor Clan
Gusai was a member of the Mantis who threatened the Emperor with a knife to his throat to force him to legitimise the Clan. He was rewarded with a Clan, a family name and his execution. The name was later rescinded as a Gusai was found guilty of attempting to assassinate a different Emperor.
Snake
Isawa Chuda was a diligent hunter of Maho-Tsukai who saved the life of the Emperor and was rewarded with land, a minor Clan and the family name of Chuda.
Sparrow
Doji Suzume was a less than competent courtier who accidentally angered his Champion, and was then used as a pawn to end the Crab-Crane Yasuki war. The land he was awarded served as a buffer and he was given the name Suzume.
Tortoise
Agasha Kasuga helped Gaijin escape the Battle of White Stag and was given a secret purpose by the new Emperor instead of punishment. He was granted a small amount of land and the right to a family name, and recruited from has Clan, the Yasuki and Gaijin survivors to form the Clan.
Wasp
Tsuruchi is the son of an affair between an Akodo and a Bayushi. His parents lived together with the Bayushi's wife in a Scorpion castle before being assassinated just before the Lion captured it. Tsuruchi later recaptured the Castle, and gained the protection of the Emerald Champion, holding it in spite of both the Lion and the Scorpion, and being granted status as a Minor Clan, but not a name.
Ronin Families
These are kind of strange: they are groups with a family name, but no official Clan title given by the Emperor
The Kaeru are a family that rule the City of the Rich Frog. They have no lords, but keep the city as a safe and neutral place for trading, mostly between Unicorn, Lion and Dragon.
A gaijin sword-smith was recognised for his skill and the 2nd Emperor declared that he could not possibly be gaijin, so he should have a family name. With no other Clan or duty, the venerable Tsi family are ronin.
Very recently (or possibly not yet?? depends on the date) the ronin Yotsu gave up his own son to rescue the son of the Emperor. He was permitted to grant his name to his children, and take a new given name - that of the Emperor's father before he became Emperor.
Lost Minor Clans
Finally, the weirdest group of people, for whom there have never been cards. These are semi-apocryphal, being described in the toolbox book 'Imperial Archives', and all with a built-in demise
Bee
Doji Hachi was a Crane courtier and art critic who gave a truthful judgement in favour of a Scorpion ikebana arrangement over a Crane one. This was at Winter Court in front of the Emperor. He was moved by her dedication to beauty, and gave her a Clan and a name - Hachi.
Crow
Mirumoto Karasu left the dragon clan to become a ronin. He gathered a small group of followers who became nomadic mercenaries that gave their wages to the Brotherhood of Shinsei. One of their number died saving a noble from bandits and the group was eventually found and given the name of Karasu and recongition as a Clan. [NB, as you can see by now, if you want to get a clan, save someone important's life]
Firefly
Shinjo Jainu was an experienced and dedicated Emerald Magistrate manning a lonely watchtower for years. When a gaijin ship was wrecked on the nearby shore his competence and quick action prevented much damage from happening before they were defeated. For a lifetime of unstinting service, the Emperor gave him the family name Hotaru and a Clan.
Salamander
Isawa Hitokage was a powerful Fire Shugenja who was defeated by a Water Shugenja at Winter Court. His subsequent outburst led to the Emperor making him a Champion of a new Minor Clan, with the family name Hitokage
Shark
An unknown samurai named Jirozame was suddenly given Imperial permission to found a Clan and use his name for the family. He and his followers acted as little better than sanctioned bandits whom the Emperor never punished. The truth is that Jirozame is a bastard son of a Hantei and a peasant.
Tanuki
A Badger yojimbo for an Imperial Magistrate became lost in the Shinomen Forest and was presumed dead. He re-appeared excatly a year later, confused by the missing time. The Emperor asked him to found a Clan to investigate the weird happenings that plague people in the Shinomen. Some people think that the entire clan were animal or trickster spirits...
Edited by Tonbo KarasuCorrections and additions