I think this is what he doesn't want:
https://www.campusquilt.com/order/
And these would be examples of what he does want:
http://www.gotquiltz.com/gallery/
It *has* to be quilted to hold the layers together, but the quilting can be quite minimal if you use Warm and Natural batting with scrim (which allows lines to be up to 10" apart).
You might want to ask him if he is okay with sashing strips. I would think sashing strips would help you achieve the "staggered" look more easily. Either way, with or without sashing, you will need to create a "map" of the finished quilt before starting so that you know how to cut the pieces and fit them together. What you might want to do is take a digital photo of each t-shirt, print them out, cut the photos into squares and rectangles that highlight the logos, then arrange the logos into a pleasing design. You can do this on top of a piece of black fabric to see how black sashing would fit. Once you get something you like, take a photo of it and get your nephew's approval. If he likes it, then re-map the whole thing onto a sheet of graph paper (4 blocks to the inch so you have 1/4" sewing increments). That will give you the *finished* dimensions of each t-shirt. You probably want to cut the actual shirts at least one inch bigger first and then, if you want 1/4" seams, cut them down to exact dimensions as you sew.
Here are a couple of tutorials I found online that might be helpful:
https://www.toocooltshirtquilts.com/...-Extraordinary
http://www.instructables.com/id/T-shirt-Quilts/