A client wants us to use this specific Uncle Sam logo:
The Uncle Sam logo was used by the US Government starting ~75 years ago. This one may or may not be a re-creation, but out of general curiosity:
If someone re-creates an old public domain icon today, do they own it, or is that piece also public domain?
If you draw your own version, it’s a derivative work, and that’s typically protected by your own copyright.
The catch is that normally with derivative work, you need permission from the original copyright holder. In this case, you wouldn’t, as it’s public domain.
All that said, as a logo, note that even if you redraw it in your own style, it’s not going to be a uniquely protectable concept. Anyone else could come along and redraw the same Uncle Sam and use it as their logo and you’d have no real recourse if there’s confusion with your client’s company.