Family Names aren't just Surnames
As Ben Barden points out in his answer, families mean something in Rokugani society. The kind of granularity you are thinking of doesn't exist, because outside of immediate family (especially the daimyo and the daimyo's children) everyone is a member of the family entire with relative positions determined by status and scope of responsibility rather than familial relations (to each other; ancestry is very important). Every family member who is not your ancestor, descendant, or sibling, is - most properly - your cousin but in a society as formal as Rokugan, that doesn't mean much.
Vassal Families are a thing, and they have different names
However, as Ben also points out there are such things as vassal families which would allow for further granularity if it is important to your campaign. Much as the Emperor has power over the creation (or abolition) of clans and the Clan Daimyo has power over the creation of families (though most often this is actually the adoption of an existing family-unit such as a minor clan or a family of another Great Clan), the Family Daimyo has the authority to create vassal families within their family. The power dynamics here are a bit more complicated than that but are not particularly relevant to answering your question. What should be noted is that any vassal family samurai is of lower status (and thus subordinate to) any samurai of comparable position in the main line. Thus a vassal family samurai of no specific account (a ji-samurai) is the lowest of all samurai in Rokugan; barely above an ashigaru peasant. (status 0.5 vs status 0.4)
Don't use Vassal Families "just because"
The point of the rest of this post is to provide examples. If you take nothing else from this post, take this:
- There are actually a fair number of vassal families, even in canon. It is not unreasonable that any given character might be a member of a vassal family, even one not mentioned in L5R canon. However, unless they are dealing with members of that Clan or hosting within their own lands, they will never refer to themselves as such (except perhaps, if it is important to the subject, in which case they may refer to themselves by saying something like "Hyuga Noman of the Utaku Family, who breeds the horses my sisters ride upon"). As such, unless you wish to a highlight a part of the campaign world or you are dealing heavily with intra-clan politics it is unlikely that their nature as vassal will be relevant.
- Vassal families always exist for a reason and always have a duty, though that duty may be loosely defined (as in the case of vassal families by award). If you're introducing a new character from a new vassal family, at a minimum, you need to defined those two things ahead of time.
- Vassal families often have traditions that vary from their "host" family, influenced as they are heavily by their founders and their duties. As such, if the vassal family is established within your campaign (a few generations old or newly adopted as a group from somewhere else) you should give some thought to what sets them apart from their "host" family. Often this will take the form of emphasis on different traits, skills, spells, etc. than those of that family normally focus on, as well as deviations in dress, rituals, and other local customs.
Background Information on What a Vassal Family is
Vassal families can be created for any number of reasons. The three most common are as an award for a singular achievement, to accommodate the adoption of an existing group into the family/clan, or to serve a specific function within the family outside the family's traditional role. Often these reasons are combined or muddled in some way.
Vassal as Award
One example of vassal family by award is the Fureheshu of the Ichiro family (Badger Clan) who's son was granted a family in Fureheshu's name after the former died, crushing an oni's skull with his bare hands.
In the case of a vassal family as an award, it may be noted that the singular achievement that earned the award (and thus the award itself) need not actually be a necessarily positive thing, with the vassal family name being to mark them as "other" from the main family. I could not easily find a specific example of "vassal family as punishment" but the Sparrow Clan provides the mold.
However, both the Byako of Kitsune (Fox/Mantis Clan) and Moshibaru of Hida (Crab Clan) provide examples of vassal families being used to contain the "other" without being punitive.
- The Byako family was first founded by a kitsune who sought to sooth the loss of the Fox Clan Champion's nephew due to an accident in Kitsune Mori. The kitsune was granted his own family, which continues with one direct decedent in each generation interbreeding with a kitsune, producing twins. One of the twins remains with the clan (to continue the line) while the other is borne away to Chikushudo.
- The Moshibaru family was founded as a place for those who succeed a Twenty Goblin Winter to learn how to be a Hida without impinging on the honor of the Hida themselves. The family consists of those who train the new samurai, and the samurai under training. Those who succeed their training and do not remain to train others are adopted into the Hida family directly.
Obviously, vassal families by award TEND to be the smallest with their very survival as a family subject to the founders fertility and/or cult of personality.
Vassals by "Adoption"
Several examples of also exist of outside groups being adopted into Clans as vassal families, such as the Chiyu of Tamori (Dragon Clan) and Kaeru of Ikoma (Lion Clan).
- The Chiyu were a group of Ronin Shugenja specializing in the Water magic and healing, lead by a former Dragonfly, who had settled on the border of Dragon and Phoenix lands. When the Tamori/Agasha split occurred, the only a handful of Agasha shugenja actually remained with the Dragon clan. In order to bolster the numbers of shungenja available to the clan, the ronin band was adopted.
- The Kaeru were a family of ronin and merchants to who ran the City of Rich Frog, in land contested between Lion and Unicorn. When control of the city passed to the Lion the Frog offered to remain administrators and so were adopted as a vassal family under Ikoma.
When groups are adopted as a vassal family it is most often because the clan wishes them to act precisely as they have always done, which may be at odds with their "host" family and thus create conflict. By making them vassal families, the clan simultaneously honors the leader(s) of the group, the nature of the group's work, and separates their traditions and their honor from that of the "host" family.
Vassals by Talent
Further examples exist of vassal families created to perform a specific task. Some examples, in brief:
- "Hunting" duty as in the case of Battue of Iuchi (Unicorn Clan) or Meishozo of Kuni (Crab Clan) who hunt maho-tsukai.
- The crafting of something important to the clan as Fundai of Kaiu (Crab Clan) who make ships or Koganshi of Shiba (Phoenix Clan) who boast the Shiba Armorsmiths.
- Specific tasks important to the family, largely ignored by the outside world, as Chukan of Asako (Phoenix Clan) who maintain the filing system for the Asako temples' records or Hyuga of Utaku (Unicorn Clan) who breed and maintain horses.
And some are just plain weird, such as the Izaku of Tamori/Agasha (Dragon AND Phoenix Clan) who serve BOTH clans as the adminstrators of the Izaku library.