Is it possible to change the line thickness used for the element edges in a wireframe/shaded view of the mesh in Abaqus?

Yes its possible to change the line thickness. Click on the Common Plot option icon (the first one first row and the one on the left to the left of the viewport).

In the dialog box which opens choose the Color & style tab. There will be 2 boxes at the bottom of this dialog box. The second one controls the line thickness of the edges of the elements in the mesh. Click on the scroll bar next to this line and choose a different line thickness.
Click on OK and this should change the appearance.