In my view, the objective of knowing the friction factor, is for one to be able to calculate what is the pressure drop needed to push a given flow $Q$ through a given pipe diameter. This kind of relations exists for several models of non Newtonian fluid, take for example the power law model:
In this case the solution gives:
where $R$ is the pipe radius and $L$ is the pipe length. You can rearrange an expression of this type to obtain an effective viscosity, depending of your definition of "effective viscosity". For example "the value of viscosity that plugged into the Newtonian pressure drop-flow relation will give the correct value of pressure drop for given $Q$".
You can review the solutions for Bingham plastic or other type of models also.
A more general approach is the one used in the Rabinowitsch-Mooney relations, where you determine experimentally a relation between flow and pressure drop, which allows you to find the shear rate at the wall, and deduce a shear rate- shear stress curve for the fluid. There are also definitions of "generalized Re", for non Newtonian flows.
Fluid dynamics books treat this topics in an accessible(algebraic, not CFD) manner, search for chapters on "non Newtonian fluids", or review Perry's Chemical engineerss handbook.
My experience is from chemical engineering though.
Hope this helps.