To robustly operate in dynamic, unknown environments, robots should be able to autonomously recover from simple errors such as tip-over. Most efforts to date have introduced specific techniques applied as point solutions on simple terrain. For a more general solution, we previously introduced a framework for analyzing and generating solutions to the self-righting problem for a generic robot. In this paper, we turn our attention toward understanding how a robot's morphology affects its ability to self-right. We begin by briefly reviewing our framework, which is used to generate the results within the paper. We then introduce a self-rightability metric that can be used to evaluate a given robot design's potential for self-righting. It can also be used to compare disparate designs. Next, we show how the metric can be used to perform a parametric study covering multiple design variables for a simple robot class. In this way, we hope to enable designers to begin to understand how design parameters such as joint limits, limb length, limb to body mass ratio, limb mass location, and body aspect ratio will affect the robot's ability to self-right on a variety of ground angles. Finally, we show a case study of limb mass and validate results using a modular, 3 degree of freedom physical robot. Ultimately, we hope to enable the production of robots that are more capable of autonomously self-righting.