The Petersen graph has an interesting 3D embedding. Take a tetrahedron. Add a midpoint to each edge. Connect opposing midpoints for a Petersen graph.
The Perkel graph or 57-cell has an interesting 3D embedding that starts with a dodecahedron. The Grünbaum graph also starts with a dodecahedron. The dual to the Klein quartic graph has a nice embedding. The various famous polyhedra and polytopes all have corresponding graphs.
The Clebsch graph can be made by adding diagonals between farthest-apart vertices of a 3D tesseract.
What other famous graphs have interesting 3D embeddings?