Why do we use Ḡ ("G bar") as a notation for partial molar free energy? For partial molar entropy there is a bar that is positioned above the $G$ with a subscript I that is sometimes added to the $G~(\bar{G}).$ The bar I read sometimes denote the "partial nature of the $G$".

However, is it not rhetorical. I am looking for the explanation as to what is so partial about this quantity with regard to it's occurrence in the equation.
Thermodynamic properties are functions of composition, which can be represented as $(\chi _1,...,\chi _N)$ where $\chi _i$ is the mole fraction of component $i$ present in the system. Now imagine a change in the composition such that the amount of component $i$ is increased by a very small quantity $dn _i$ (alternately, imagine the system to be infinitely large), and a thermodynamic property $X$ is changed by $dX$. Then (using the notation you describe) we define the partial molar property as $$\bar{X}=\left(\frac{\partial X}{\partial n_i}\right)_{p,T,n_j}$$ This notation means that we make a limiting, infinitely small change in $n_i$, such that the change in $X$ is for all practical purposes nil, ie $\Delta X=0$. The partial molar quantity describes the slope of the property $X$ (which can be represented as a surface in a multidimensional manifold) along the composition coordinate $n_i$, at a specific composition $(\chi _1,...,\chi _N)$.