To solve this problem, we need to understand the symmetrical properties of a cube and how a rotation about one of its body diagonals affects its appearance.

First, let's identify the body diagonals of a cube. Each body diagonal is a line segment that connects one vertex of the cube to the opposite vertex. There are four such diagonals in a cube.

When the cube is rotated about one of these body diagonals, the rotation that leaves the cube looking the same must map vertices to vertices in such a way that the spatial arrangement of all vertices remains unchanged.

A cube has rotational symmetries corresponding to rotations of $$120°$$ and $$240°$$ (in addition to the trivial rotation of $$0°$$) about any of its body diagonals. These angles are derived because a body diagonal rotation permutes four vertices around the diagonal in a cyclic manner, as each vertex can map to one of three other vertices after a $$120°$$ turn, and so forth.

To formally describe this, if we fix a body diagonal and rotate the cube:

A rotation of $$120°$$ will cyclically permute three vertices symmetrically around the diagonal.
A rotation of $$240°$$ is simply $$120°$$ in the opposite direction and also keeps the cube looking the same.

Therefore, the minimum angle of rotation about the body diagonal that leaves the view of the cube unchanged is $$120°$$.

Thus, the correct answer is:

Option A: 120°