What is the singular for an octadic?

With your picture, I wonder if octile is not simply what you are looking for?

Quarter => quartile
Eighth => octile
Tenth => decile
Hundredth => percentile

For the specific diagram you give, an eighth-part is an octant, using the following senses of the word.

1. The eighth part of a circle; an arc of 45 degrees.
2. (geometry): The eighth part of a disc; a sector of 45 degrees; half a quadrant.

A fourth of the diagram is a quadrant, in the sense “One fourth of a circle or disc; a sector with an angle of 90°”. (More commonly, quadrant has the sense “One of the four sections made by dividing an area with two perpendicular lines” or of being one of the regions of the Cartesian plane that are bounded by the x-axis and y-axis; the “first quadrant” being the region where x>0, y>0; “second quadrant” where x>0, y<0; etc.)

A single section of 8 would be an eighth (notice the extra "H" at the end):

eighth [eytth, eyth]

adjective

1. next after the seventh.
2. being one of eight equal parts. noun
3. an eighth part, especially of one (⅛).

But as oerkelens says, a single of an Octennial is a year.