The Doctor Herself?
A better explanation for who is the timeless child is that it's the Doctor herself. When the Remnants are reading her mind, they're searching the Doctor's mind for her lifetimes of fears, and it's only when they dig far enough back that they uncover the timeless child. This would suggest that the Timeless Child is the Doctor as a child, before becoming a Time Lord and certainly before any regenerations.
There's very little we know about the Doctor's childhood, but they're often described as being lonely and isolated as child. In the one episode where we actually see the Doctor as a child (a Twelfth Doctor episode, "Listen"), they're afraid of the future that awaits them, unsure of going to the academy and becoming a Time Lord. Perhaps this means that Timeless Child is an insult: a phrase used to describe a young Gallifreyan who isn't suited to becoming a Time Lord. It'd make sense, then, that this is a period of her life that the Doctor would rather not remember, likely having suppressed these memories and therefor having hidden it from herself.
If this is the case and Timeless Child refers to the Doctor's childhood, then it seems all but certain that season 11 will further explore the Doctor's childhood - maybe even revealing how the Doctor became a Time Lord. It may also explore why it is that the Doctor is so unlike other Time Lords, since the Doctor's curiosity and want for adventure is considered unusual.
A Brand New Character?
Of course, while there's a good chance that the Timeless Child is a reference to either the Doctor's childhood or some offspring of the Doctor, it's just as likely to be something entirely new for the series. Doctor Who season 11 has made a point of being accessible for new viewers and including something so intertwined with the Doctor's history wouldn't fit that mission at all. This may be the least interesting of possible answers, but it's also the one with almost limitless possibilities. The timeless child could be another person, or a folk legend, or maybe even a condition that can infect Gallifreyan children.
This early on it's nearly impossible to guess what the Remnants meant by timeless child and why it has the Doctor so upset. But what is certain is that the phrase and the meaning behind is going to be an important mystery for Doctor Who season 11 to unravel.
Next: 10 Big Questions After ‘The Ghost Monument'
Doctor Who season 11 continues next Sunday with "Rosa" on BBC One and BBC America.