L.P.
You can definitely reattach the cut out name to another shirt...
Go to a fabric store and buy some fusable webbing. It's a whitish/clear film that is paper-backed on a roll... you iron the webbing, paper still on, onto your cut out design. Then peel the paper off, lay the design with webbing now ironed onto it, on the new shirt, then iron again to attach it to the new shirt. You are essentially creating your own Iron-On design by attaching the fusable web to your design. It works like a charm. I've done it with lots of stuff. There are better directions on the paper attached to the webbing, but I gave you the nutshell version!
Another, slightly more complicated, but really cute thing you could do, is use a second fabric in your son's team colors... say, if the team is red and white, and the team name you'll cut off your shirt is red and white, get a piece of printed fabric in those colors, maybe a plaid, or polka dot, or check pattern, slightly larger than your design... cut our your design in the shape you desire... then use your design as a guide, and cut out the same shape in the other print, making it about 1/4" bigger than your designs on all sides, and create a border of sorts for your design! You'd use the fusable webbing to apply the printed fabric to your new shirt first, then apply the design to the printed fabric on the shirt, using the same steps above.
Tip: If you cut your design larger than you'll need it, and attach the fusable webbing to it before trimming it to the exact shape you desire, you'll get a nicer, cleaner cut edge, as the webbing adds structure to the fabric (more like cutting paper than fabric once the webbing is attached.) You won't get frays like you would if you cut the fabric to your exact shape first.
If you do it this way, you will probably have a clean enough edge that you won't even need to add a trim. When I've done this, I like to trim both the printed fabric, and the design with a thin line of fabric paint, or puff paint, just to finish it off.
If you would like more detail, just pm me! Good luck!