The catch-all terms in your list belies the differences that exist. There are radically different kinds of eyes. For example, your eye, insect eyes, octopus eyes, and lobster eyes are all wildly different. The same is true with lungs (invertebrate lungs vs vertebrate lungs). This is also true with hearts, limbs, ears, mouthparts, and taste/smell organs
For the eye, specifically, it would not be surprising if we were unsure whether it can be evolutionized to a single initial version of a primitive eye. The issue is that if you go far back enough we just aren't sure due to lack of fossil evidence. The simpler things are, the less variation there can be and the more likely two independent things will end up the same. If there were a common origin for all eyes, it would likely be a light sensitive organelle on the microorganism level and you can see why fossil evidence for such a thing is tricky to discern even if found it.
Even going as far back as the mysterious Pre-Cambrian may be insufficient. Jellyfish, molluscs, and worms (our ancestors) already existed, and likely arthropods (marks that look like leg tracks have been found), all of which are known to be capable of light sensitivity...in modern times at least.
But let's not forget plants can detect light too and they go back even farther so that could be your evidence for convergent evolution of eyes...but again, it's also not a sure thing unless you can find and compare genetic evidence for light sensitivity between living plants and animals).
But eyes are a difficult and unnecessary example if you just want completely independent convergent evolution examples. Legs are dead easy. Arthropods and "fish" diverged before the "fish" even had fins (from which limbs would evolve for all tetrapods), and arthropods already had legs.
Another easy one is wings between insects, bats, pterosaurs, and birds. Less "optional" structures such as internal organs (and especially body symmetry) tend to go back really far and make things difficult.