Quote by Matterwave
The set of coordinates on the initial hypersurface has some origin somewhere.

A coordinate system doesn't have to have an origin.
Quote by Matterwave
Is this origin carried to the subsequent hypersurface by this vector t or by the normal vector to the hypersurface? Meaning, if I started at the origin and move along this vector t to the next hypersurface, do I stay at the origin, or do I move away from the origin of coordinates? Or is this arbitrary based on how I define my coordinates?

It depends on the coordinate system.
Quote by Matterwave
I'm confused here because intuitively to me it seems like I would prefer to have my coordinates move along with the normal

It is more natural to choose a coordinate system that consists of [itex]t[/itex] and spatial coordinates that are constant along the integral curves of [itex]t^\mu[/itex].
Quote by Matterwave
but MTW makes a statement that the shift vectors tell you where these "normal struts" end up on the above hypersurface via the equation:
[tex]x_{new} ^i=x^iN^idt[/tex]
Which seems to suggest that these "normal struts" (normal vectors) do not start and end at the same coordinates.

Because they don't start and end on the same integral curve of [itex]t^\mu[/itex].