Visualizing Unique Games
If I were going to illustrate unique games, I'd do something with a label-extended graph (similar to the suggestion you mentioned).

But specifically, I'd contrast the original constraint graph with the label-extended one. For instance, label edges in the constraint graph with the corresponding equation out of:

$x_1 - x_2 = 0\mod 3$

$x_2 - x_3 = 0\mod 3$

$x_3 - x_1 = 1\mod 3$

Then there will be a (hopefully visually appealing) "twist" between the $x_1$ and $x_3$ vertex clouds in the label-extended graph.