Pre-ordered my copy of Mass Effect: Andromeda the other day, and already coming up with canon links between it and the origtrig.
I always have my Shepard romance Liara, so my primary canon is gonna be that Peebee is the love-child of Liara and Shepard that she never tells him about. Here goes………
The relationship is built up through the course of the first game, beginning (as stated by Liara) as a simple, scientific intrigue, and growing into something much greater. On their way to Ilos, the two finally join, and Liara becomes pregnant. Two weeks after they take down Saren, Shepard is spaced and the crew disperses. It is still too early to show any sign of her carrying, though, and Liara goes off to start her own life up again. She has the child in secret, and hides her away when she goes out gallivanting, looking for Shepard's corpse and becoming an information broker, only seeing her and her adoptive family periodically from then on.
When the two lovers meet again on Illium, Liara sees that Shepard is under a lot of stress and couldn't handle the knowledge of his daughter being out there on her own, or think that he abandoned her. She says that "so much has happened", but it's nothing she's willing to discuss in the present moment. The rekindling of the romance during their pursuit of the Shadow Broker warms her up to the thought of telling Shepard about their daughter, but he's never around - off saving the galaxy again - so she puts it on the back-burner while she manages all that the Broker left behind.
It's finally just before the Reapers hit Earth - while Liara is working in the Mars Archives, so close to Shepard - that she's ready to tell him about their daughter over vidcomm (even though she's already been adopted by another well-to-do Asari family, recruited by the clandestine Andromeda Initiative). Liara knows this, and wants Shepard's mind to be at peace and focused before he's thrown into weirdness again; that he has a daughter out there, counting on him, would be allthemore of a bolster to his morale. Again, it gets swept up in the confusion, though, and the daughter of Shepard is but a memory of Liara, and a figment of Shepard's imagination - a now-unattainable goal that he doesn't even know exists.
The daughter of Shepard goes her own way, knowledgeable of her father, missing her mother, forever separated from her home by millions of light-years.