Depending on the type of nanoparticles you want to investigate many important properties of the nanoparticles are influenced by their polydispersity. I'll give you some examples:
The catlytic activity of noble metal nanoparticles is influenced, due to the changes in the surface to volume ratio.
Article The Impacts of Nanotechnology on Catalysis by Precious Metal...
The formation of colloidal crystals by hard sphere nanoparticles can be suppressed, if the polydispersity of the nanoparticles becomes too high.
https://www.nature.com/articles/35099513
In terms of biophysical properties, the biodistribution of nanoparticles is influenced by their size, so polydispersity is an important parameter for drug delivery systems.
Article Nanoparticles for drug delivery: The need for precision in r...
From an analytical point of view, a high polydispersity of your system will smear your data, if you use small-angle scattering methods to characterize them which makes the data sometimes difficult to analyse.
Article Polydispersity Effects in the Small-Angle Scattering of Conc...