As shown from the Snicket family tree, H. is the child of D. Snicket and the (presumably middle, seeing as their name is listed second both on the tree and alphabetically) sibling of G and I. They are the grandchild of Chas. Snicket, the grand-nibling of A and B, the nibling of E and F, and the cousin of Jacques, Kit, Lemony, M, N and O. This would make Beatrice Baudelaire II their first cousin, once removed.
H's siblings, G and I, are often theorized to be Gregor and Ike Anwhistle, which would make H a member of the Anwhistle family. As many VFD families are interconnected, it would make sense that the Snickets and Anwhistles were related. This could also make H's parent, D, his mother, who was described as having one eyebrow and one ear.
If H. is an Anwhistle, that would make them the second cousin of the Baudelaire family.
In addition to this theory, H has been theorized to be Hector, who is known to be a member of VFD (like the Snickets and Anwhistles), and whose surname was never revealed. Jacques also refers to Hector as "H" in his letter to Lemony, mentioning Hector's ballooning interest.
A very small disadvantage to this theory is that when Hector talks to Lemony in All the Wrong Questions, any relation between them is never discussed, though they are also talking about Kit, though this could easily be refuted- only part of their conversation was shown, and they had more pressing matters to discuss. A much larger disadvantage is that Hector never mentions any relation to Jacques in The Vile Village, when Jacques is in front of him at the townhouse and then murdered, however this could also be excusable considering that Hector was afraid of the town council, and seemed to be actively hiding his previous involvement with VFD.
|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|