There are many parents that give one of their children, typically the first child, the same name as their father, and so they are known as John Junior, and then if that tradition continues, you will see a John III and John IV and so on. Those that are named after their father may be known by that given name throughout their life, whereas with others, via by default or by choice they are known via a nickname, or perhaps they utilize the first initial of the first name with their middle name, or even just those two initials, or perhaps just the middle name. In any event, the commonality of sons being named after their fathers is well established in America.
On the other hand, it seems, it is truly exceptional for any daughter to be named after her mother, so that we seldom see a Jane Junior, although this does occur occasionally, perhaps at a ratio of 1:100 to males, but it does happen. What is puzzling is if to be named after the father is good enough for the son, then the naming of a daughter after her mother, should be accorded the same courtesy. That is to say, what is good for the goose should also be good for the gander.
In point of fact, the naming conventions of girls are always going to be a bit more problematic mainly because most females upon marriage give up their last names so that Jane Smith Junior would probably be known simply as Jane Jones, and the Junior designation would therefore be dropped. Because of this there is now presented also the conundrum of not being able to take a female Junior and being able to successfully pass that same name down to the next generation as, for instance, Jane Smith III, although in actuality, just because the last name has changed, doesn't necessarily mean that it can't be done.
Further, you can make a very strong argument especially in today's world, in which some women despite getting married never give up their last name, or instead combined their last name with their spouse, that the time has never been better for women to follow, if so desired, naming conventions which primarily had been utilized just for the male side of the family, and use it as their own, so that there can be multiple generations of the same female first name as well as the last.
The thing is with more and more women being quite accomplished in their own right, and celebrated for their achievements -- that the naming of a female child after the matriarch of the family, seems like a reasonable thing to do for those that have an affinity for this sort of honor and tradition within a family. It just seems, in general, that those that name their child after themselves, are in a way, testifying that their reputation and that their name is one that deserves to live on in the new incarnation of their child. If this then is true, and the desire is also there, then it just seems logical to pass that same tradition onto the female side of the family, because they deserve the same opportunity as the males.