I want to say a couple of things regarding the self-proclaimed Whitebeard's beloved and her 'son'.
No matter how you look at it, she cannot be Whitebeard's "lover".
I am certain her appearance is also being used as a 'gag' by Oda-sensei.
Whitebears could not fall in love with these kind of women. A woman who tramples over Whitebeard's ideals of what 'family' is cannot be loved by Whitebeard.
She appears to be a witch in both the moral sense of the word and the literal sense of it.
She appears to be using this whole Whitebeard's lover and Weeble's mother shebang so as to collect 'money,' especially the 'bounty money' on the head of Whitebeard's sons. She has shown great longing for money.
When he destroyed the 'AO Pirates', she told him to tell her about their 'next assignment', which is the annihilation of 'Marco the Phoenix'.
This implies that the 'previous assignment' was what had just happened. In other words, the annihilation of the 'AO Pirates.'
This whole Whitebeard's-son shebang appears to be connected to Whitebeard's "vast inheritance".
This is what appears to be her ultimate objective from this entire act.
As for Weeble, he truly believes that he is Whitebeard's son insofar as he was shedding tears of anger over his father's death and swearing vengeance on Blackbeard, the one who committed this patricide.
He does not appear to be having a hand in this 'act'. He appears to truly love Whitebeard.
She also appears to be manipulating him using a picture of Whitebeard to help sustain this entire charade.
I don't know about the authenticity regarding the notion that she is his mother, but I can tell that there is no love in her acts. What kind of loving mother would be using lies and deceit to gain profit from her child's acts?
He appears to be similar to him in many aspects: the bisento, the mustache, and, according to Kizaru, the strength
I don't think he is Whitebeard's son. However, it is possible to have Whitebeard's blood running through his veins. He could be some sort of clone made using some sort of sorcery or something like that. His proclaimed mother looks like a 'witch' after all.
EDIT: Weeble and his mother remind me of Majin Buu and Babidi(from Dragon Ball Z).