What is the complexity of this path problem?
This problem was studied in:

Shiri Chechik, Matthew P. Johnson, Merav Parter, David Peleg: Secluded Connectivity Problems. ESA 2013: 301-312.

http://arxiv.org/pdf/1212.6176v1.pdf

They called it secluded path problem. It's indeed NP-hard, and the optimization version has no constant-factor approximation.  

The motivation the authors provide is a setting where the information is sent over the path, and only the neighbors and the nodes in the path can see it. The goal is to minimize the exposure.