For a liquid electrolyte-based dye-sensitized solar cell (DSSC), long-term device instability is known to negatively affect the ionic conductivity and cell performance. These issues can be resolved by using the so called quasi-solid-state electrolytes. Despite the enhanced ionic conductivity of graphene nanoplatelets (GNPs), their inherent tendency toward aggregation has limited their application in quasi-solid-state electrolytes. In the present study, the GNPs were chemically modified by polyethylene glycol (PEG) through amidation reaction to obtain a dispersible nanostructure in a poly(vinylidene fluoride-co-hexafluoro propylene) copolymer and polyethylene oxide (PVDF–HFP/PEO) polymer-blended gel electrolyte. Maximum ionic conductivity (4.11 × 10−3 S cm−1) was obtained with the optimal nanocomposite gel polymer electrolyte (GPE) containing 0.75 wt% functionalized graphene nanoplatelets (FGNPs), corresponding to a power conversion efficiency of 5.45%, which was 1.42% and 0.67% higher than those of the nanoparticle-free and optimized-GPE (containing 1 wt% GNP) DSSCs, respectively. Incorporating an optimum dosage of FGNP, a homogenous particle network was fabricated that could effectively mobilize the redox-active species in the amorphous region of the matrix. Surface morphology assessments were further performed through scanning electron microscopy (SEM). The results of rheological measurements revealed the plasticizing effect of the ionic liquid (IL), offering a proper insight into the polymer–particle interactions within the polymeric nanocomposite. Based on differential scanning calorimetry (DSC) investigations, the decrease in the glass transition temperature (and the resultant increase in flexibility) highlighted the influence of IL and polymer–nanoparticle interactions. The obtained results shed light on the effectiveness of the FGNPs for the DSSCs.