Why are root hairs mostly unicellular?
A couple reasons come to mind.* The purpose of root hairs is absorbtion. If you consider multiple cells along the length of the hair,  additional cell walls and plasma membranes would somewhat impede the movement of nutrients.

More importantly, the absorbtion is a function of the surface area of the hair. Smaller diameter (one cell thick) hairs provide greater surface area to volume ratio. While it's true that a larger diameter, due to multiple cells, might offer greater overall surface area, it would come at the expense of requiring more volume compared to a greater number of smaller hairs.

For example, suppose you have a certain volume of cutoplasm to spend on making root hairs. For the sake of the example, suppose the hairs are of a fixed length. If you make the hairs half the diameter you will be able to provide twice the overall surface area using the same volume of material. 

* By this I mean I don't know the ultimate (evolutionary) reason and I doubt anyone does. This is probabably not an active area of research and you're not likely to find definitive answers with research. So, my answer, if you will, is conjecture. 

That said, it is educated conjecture. In 24 years of teaching AP Biology, the relationship of volume versus surface area has arisen multiple times. One of the exercises I gave my students, in preparation for the exam, was to find at least ten examples. Some of the best students found many more.