Obviously, these are huge spoilers for the original series:
My understanding is that Unit 01 is actually the first EVA unit created. It was quite literally pulled from Lilith (the angel they have crucified down in the basement -- that's why they only have the top half of her; you can see this in process in some quick scenes flashing back to the original Contact Experiment). The initial plans and intentions are unclear, like a lot of things in this series, but after Unit-01 absorbed Yui Ikari, Unit-00 and Unit 02 were created in a different manner, from cloned cells from Adam (the First Angel who blew up in Second Impact). It's probably better to say that Unit-00 was the first EVA unit that was functional, and it's the testbed for the control and containment technology to keep the monsters biddable, like the plate armor and the giant crucifix they jam into the spinal column (termination plug). Unit 01 was armored and armed after that testing had taken place, and given the designation then; Unit-02 was then built as the "production" model.
As to why Shinji is then the Third Child, and not the first, well, it's certain that he was not known, to the public or to the Human Instrumentality Committee, to be a pilot before Gendo summoned him to Tokyo-3. It seems possible that Gendo did not intend to have Shinji pilot at all, meaning to run the whole system with Rei until the Dummy Plug was operational -- but when the Third Angel was about to attack with Unit-00 and Rei disabled, he grabbed Shinji to pilot, knowing that Unit-01 would sync with him because of the soul inside it. It makes sense that the "public" pilots would have designations given to them first.
Now, extra-textually, you could also consider some symbolic considerations. Rei is first, but she's rei (zero), without a soul of her own, so she pilots Unit 00. Asuka is all about duality -- everything the current 'tsundere' trope is -- so she is the Second Child and pilots Unit 02. Shinji -- well, he's the Third, and he pilots Unit 01 -- and when three are really one but still three, Christianity calls that the Trinity, and so Shinji ascends to the Godhead in Third Impact.
(This is extreme armchair symbolism and very basic, so take it with as many grains of salt as you need.)