Does there exist a dye that will primarily bind with water in a water/oil emulsion? I'm wondering if there exists a type of dye that will bind primarily with water when introduced to a water/oil emulsion. That way if the water is centrifuged to the bottom of the vial, it will be dyed showing better visibility for the meniscus line.

The main application of this would be for centrifugation of clear oil that has suspended water. If the oil is clear and spun out, the water may separate but seeing the distinct meniscus can be difficult. If a dye in introduced and binds to the small water content within this emulsion, once it's spun out, ideally you'd see the dyed water at the bottom. The content of water is probably 0.5 - 1.0%

I'm also somewhat aware of some water by distillation methods using dyes to detect small amounts of distilled water, but I'm unfamiliar with these and I can't find any helpful resources online about it.

I suppose it would need to be a hydrophilic dye that also resists binding to hydrocarbons?
You are asking for water-soluble dyes. Typical applications are dyeing methods in microscopy and food coloring.

Water-soluble dyes for microscopy: e.g. Eosin Y, Methylene Blue, Ethyl Green.

See also this table: Wikipedia in German: Lebensmittelfarbstoff - Deutschland.