I modeled a new part in 3ds max using calipers and the broken part as a reference. The model was split in two and laid flat to ease printing without support. This orientation is also the best for strength in this case, as there is little forces prying the layers apart. Since the white pin keeping the lid and ring to the hinge was still okay, it was reused. The two halves were glued together lightly, not that this was completely necessary as it would be kept in palace by the pin and the hole in the toilet.
The part was printed in PLA with pretty high infill, and it felt very strong. However it lasted only about a year or so before the brittle plastic failed to repeated use and impacts. Now the question was raised again, is it perhaps best to just buy a new lid and forget about trying to do improvements to the print? Even with 100% infill it did not feel like PLA or ABS was up to the task, the existing part was made of a much softer plastic capable of withstanding higher forces even if it did bend quite a lot.
I was not one to give up 3d printing just yet, so I thought to test the part printed in nylon which was a new material to me at the time. Nylon being very tough and bendy like the original material for this part. It was a great success, and the part is still in use more than 3 years later.