Your grandfather's niece is the first cousin to one of your parents. That would make you first cousins once removed. Most families just use the term cousin.
First cousin once removed.
Most likely they would not be related to you
Your grandfather's niece is a first cousin of one of your parents. Her son is your second cousin.
Your grandfather's sister is your great aunt.
What is my godfathers niece to me
For clarity, lets call your great great grandfather's niece "the niece". The relationship sequence: (1) The niece is the daughter of your great great grandfather's sibling (2) The niece is a 1st cousin of one of your great grandparents (3) The niece is a 1st cousin once removed of one of your grandparents (4) The niece is a 1st cousin twice removed of one of your parents (5) The niece is your 1st cousin three times removed
Your niece's brother is your nephew.
cousin
Your son and your niece are first cousins.
3rd cousin
Nothing. Your niece in law. Your brother in laws niece.
Your great great grandfather.
cousins