Changing a name might seem like a minor matter for those who are changing their gender identities and, for some, facing challenges like finding knowledgeable doctors, trying hormones and experimenting with painful hair-removal procedures. But many who have gone through the switch say a name change sends an important message to the world, a message solidified and made official with a court’s approval. In many courts around the country, what were once risky or shocking name-change requests are becoming more routine as the sting of gender taboo has lost a little of its edge. But in few places has this shift been more dramatic than in New York, where two recent and little-noticed rulings helped clarify the murky area not only of the law but also of modern gender identification. They have contributed to Manhattan’s becoming a capital of Joe-to-Jane proceedings. A rare network of some 200 lawyers now works on such cases filed in the Centre Street courthouse, and nearly 400 of their transgender clients so far have, more or less, become someone else.