The child who we see with Silvia Christel after the "real" ending of No More Heroes' credit roll, dubbed Jeane, is one of the most speculated elements of the game. Some assume she is the daughter of Silvia and Henry, but no one can be completely certain at this time. Her hazy origins made me feel as though her inclusion was simply meant to be another way for Goichi Suda's script to blow the player's mind, before curtains closed on No More Heroes for good.
I've been frequenting the GameFAQs Message Board for No More Heroes 2: Desperate Struggle lately, where the user Pyrodragon20 wondered about the significance of the name "Jeane," which appears three times in No More Heroes (an assassin, a cat and the child who this blog is focusing on). It is my opinion that this naming convention started with the assassin, also Travis Touchdown's long lost love, who later steps back into the picture and reveals to us that she is actually his half-sister. Touchdown, unaware of the relation, keeps a picture of the woman on his desk to remember her by, and called his cat Jeane so she could be the namesake of his love.
As for the child we see with Silvia, anything could be possible, but this is my interpretation. Through flashbacks during his conversation with Jeane (the assassin), we learn that the night before the game's events begin, Travis went to the bar and got hammered, which brought the innate memories that Jeane is his parents' killer to light (he doesn't seem to remember while sober until Dark Star helps him with the revelation). At the bar Travis babbles about how Jeane killed his parents, and Silvia tells him that she can help him to get even. Back to reality, Jeane declares, "So you and I finally meet. You should thank Silvia." This statement, along with Silvia's words at the Death Match bar, gave me the impression that Silvia and Jeane actually know each other, and this acquaintanceship is how Silvia planned to help bring Travis face to face with his parents' killer. However as a con artist (or so we're being led to believe), she still took it upon herself to embezzle him through a series of challenges with other assassins. Travis realizes upon meeting Jeane that "They were all for this." Anyways, Jeane is also Henry's half-sister, but we're never given any reason to believe he knows about his father's past family (after all, Travis didn't), so we don't have a lot of evidence to prove the child Jeane (if she is in fact his and Silvia's daughter) was named after his sister. We can however assume that Silvia and assassin Jeane are somehow acquainted, so the child Jeane may have been named after Silvia's friend, who as fate would have it turned out to be her own husband's half-sister.
That is my best guess as to how the girl came to be named Jeane. As for who she is exactly, only Suda knows. I read someone's opinion not too long after the game was released that the child could be the reincarnation of the assassin Jeane. Personally I offered some speculation of my own, but as things stand it's completely unverifiable, as the only thing we can confirm about her now is her name. Still, I tried using symbolism within the game to draw some connections.
As far as I understand it, in Flower, Sun, and Rain, Sumio Mondo is a clone of Sumio Kodai. I'm an English speaker so I can't play The Silver Case to learn any other possible information about the situation (I've read that Mondo actually is Kodai, but Kodai is Japanese whereas Mondo is supposedly from the Shelter, located in Micronesia; he also seems younger and has no recollection of Kodai's life, even when he meets Kodai's previous partner). Anyways, whether or not Mondo is a clone of Kodai, we know Mondo himself went on to be cloned over a dozen times. With that knowledge, my speculation is that the cloning theme may make a reappearance in No More Heroes, with the child Jeane. Silvia, who knew the original Jeane before her death, could have taken it upon herself to raise this new incarnation of her friend. I noticed a connection between the child Jeane and the assassin Jeane in her youth that they are both seen wearing white dresses (in the case of the assassin, I'm referring to the photograph on Travis' desk).
If the child actually is intended to be a clone or some other sort of incarnation of Jeane, this could greenlight her re-appearance in the future. It would be difficult to say in what manner though, because Jeane decided in the first game that she'd "had enough" of life, so being cloned might not have been in her best interests. But at the same time, just before this decision she begs Travis not to kill her, so whether she would be up to coming back for more is anyone's guess really.
Most of the text from this interpretation was reworked from two postsI originally made on the GameFAQs Message Board on August 18, 2009.