Is there an accepted rule for naming all of our various distant relatives (kinship terms)?
Your cousin's first son is your first cousin once removed. It is quite confusing!
It's very easy to determine the relationship in English.
First, find your common ancestor. Siblings have the same father, cousins have the same grandfather, etc.
- Same grandfather means First cousin.
- Same great-grandfather means second cousin.
- Same great-great-grandfather means third cousin.
If you are not in the same generation, pick the shortest one, then the generation distance is how many times removed.
- Your cousin's child is your first cousin, once-removed.
- That person's child is your first cousin, twice removed.
The labels work the same in both directions so "first cousin, once removed" identifies the relationship to the speaker irrespective of whether that person is in the younger generation or not.
My brother is into genealogy, and he described for me a simple way to remember these relationships.
Going up-generation uses ordinals. So your cousin is your first cousin. Your parent's cousins are your second cousins. Your grandparent's cousins are your third cousins.
Going down-generation use removes. So your first cousin's child is your first cousin once-removed. Your third cousin's grandchild is your third cousin twice-removed.
This contradicts Mr. Shiny's answer though, because the relationships are not symmetrical. To my second cousin (my father's cousin), I am a first cousin once-removed.