Using biomass to develop and obtain environmentally friendly and industrially applicable biomaterials is increasingly attracting global interest. Herein, cellulose nanocrystals (CNCs) and lignin nanoparticles (LNPs) were extracted from Lemna minor L., a freshwater free-floating aquatic species commonly called duckweed. To obtain CNCs and LNPs, two different procedures and biomass treatment processes based on bleaching or on the use of an ionic liquid composed of triethylammonium and sulfuric acid ([TEA][HSO4]), followed by acid hydrolysis, were carried out. Then, the effects of these treatments in terms of the thermal, morphological, and chemical properties of the CNCs and LNPs were assessed. The resulting nanostructured materials were characterized by using Fourier-transform infrared (FTIR) spectroscopy, X-ray diffraction (XRD) spectroscopy, thermo-gravimetric analysis (TGA), and scanning electron microscopy (SEM). The results showed that the two methodologies applied resulted in both CNCs and LNPs. However, the bleaching-based treatment produced CNCs with a rod-like shape, length of 100–300 nm and width in the range of 10–30 nm, and higher purity than those obtained with ILs that were spherical in shape. In contrast, regarding lignin, IL made it possible to obtain spherical nanoparticles, as in the case of the other treatment, but they were characterized by higher purity and thermal stability. In conclusion, this research highlights the possibility of obtaining nanostructured biopolymers from an invasive aquatic species that is largely available in nature and how it is possible, by modifying experimental procedures, to obtain nanomaterials with different morphological, purity, and thermal resistance characteristics.