As shown on the Snicket family tree, O. is the child of F. Snicket, and the (presumably youngest, seeing as their name is listed last both on the tree and alphabetically) sibling of M and N. They are the cousin of G, H, I, Jacques, Kit and Lemony. This would make Beatrice Baudelaire II their first cousin once removed.
One theory is that O is Olivia Caliban, and M is Miranda, as they are both close relatives; as there are many interconnected VFD families, it would make sense for the Calibans and Snickets to be related. This would also make Friday Caliban part of the Snicket family tree, as Miranda's daughter. This theory is presumably non-canon in the TV series, as Olivia and Jacques Snicket (who would be her first cousin in this theory) are romantic interests, but it is still plausible in the original Book canon.
Another theory is that M, on the family tree, is Monty Montgomery, which could make O Monty's sister. Outside of the initial and the fact that Monty has at least one sibling, there is not much evidence for this theory.
Another theory is that O is Count Olaf; this is also sometimes tied to the Montgomery theory, as Olaf's surname is never revealed. The initial "O" almost always refers to Olaf in the series, and his tendency towards wicked deeds could explain the noose hanging from the O branch in the family tree illustration. However, it is very unlikely that Olaf is such a close relative of the Snickets, considering his romantic relationship with Kit.
|A. Snicket||B. Snicket||Chas. Snicket|
|D. Snicket||E. Snicket †||Jacob Snicket †||F. Snicket|
|G. Snicket||H. Snicket||I. Snicket||Jacques Snicket †||Kit Snicket †||Lemony Snicket||M. Snicket||N. Snicket||O. Snicket|
|Beatrice Baudelaire II|