They would be your step-grandchild.
The English language does not have a recognized term for the husband of a niece. Most often, when talking about him you would say, "my niece's husband" or " my sister's (or brother's) son-in-law."
Your niece in-law, or your niece by marriage.
If you are a female you could be talking about yourself. You could also refer to a sister this way. This could also refer to a cousin of yours, since your aunt's husband may have nieces through more than one sibling (either his siblings or your aunt's siblings). It is not clear whether you are related to your aunt or to your uncle by blood, but it is very probably one of them and not both.If your aunt is the sister of one of your parents, then her husband (your uncle) may have nieces who are not related to you at all. These would be the daughters of his siblings.
The term "great niece" or "grand niece" is a parallel construction from the more common terms "granddaughter" and "great granddaughter."
Your niece's brother is your nephew, not her husband.The English language has no special term to for the husband of a niece. If you want to refer to him without using his name, the best you can do is "my niece's husband," or "my sister's (or brother's) son-in-law."
Your niece's son is your great-nephew. Your niece's grandson is your great-great nephew.
Your niece's daughter is your great niece and her granddaughter is your great great niece. You are the granddaughter's her great great aunt or great great uncle.
Your niece's son's daughter is your great grand niece. Your niece's son is your grand nephew or your great nephew, whichever term you prefer.
Your brother's daughter is your niece and his granddaughter is your great niece.
Your daughter and your niece are first cousins. Your daughter and your niece's daughter are first cousins, once removed.
You would be her aunt. She would be your niece.
Your sister's daughter is your niece and you are either her aunt or her uncle. The English language has no term for your niece's husband, other than calling him "niece's husband."