Probably through the production or consumption of biological pigments.
I doubt that the genes responsible for melanin production in humans, and therefore eye color, could easily mutate to produce a red pigment. You would need to postulate an entirely different gene. That is to say, we are in the world of speculative fiction here, not realistic human biology.
However, that said, there are various possibilities. One of the most common classes of red pigments in nature is a group know as the carotenoids. These are quite common; however, very few animals are capable of synthesizing them. Most animals that use these as coloring must first consume them, whether in plant or animal form. So someone using this method would have to eat a lot of carrots.
There are some other types of red-orange pigments that most likely do not require dietary pigment intake, for instance the psittacofulvins of parrots.
Either of these pigment classes could give a human red eyes, if they had the appropriate genes. However, carotenoids are probably a superior method. The only real red-eyed species of which I am aware are amphibians (frogs in particular).
One of the most extreme is of course Agalychnis callidryas, shown below. It is very likely that the red-pigmentation is due to carotenoids, since other frogs use carotenoids to color their skin, at the very least. Note that the tree frog does not have red skin, so skin and eye pigment production are clearly somewhat separate.
So there are definitely proteins in nature that can be used to color eyes red, with at least one probable proof-of-concept in the carotenoids. I don't think typical baseline-human mutations will cover it, though.
How would it be diagnosed?
Well, removing the person's eyes and subjecting them to chemical tests might allow determination of the pigment with 1910s technology. Obviously, this would not happen. The genetic basis of inheritance was already well-established by then, and indeed the nature of chromosomes as gene carriers was known. Without detailed knowledge of biochemistry, all that the physicians could determine would be that it was congenital. In reality, of course, even the pigment could not be determined, since the very best optical microscopes have a resolution of perhaps 200 nm, and electron microscopy would have to wait until the 1930s at the earliest. Something like beta-carotene is 1.9 nm in length. That is something of a long molecule.
So without taking out the eyes, there would little chance of figuring out that carotenoids were involved. Probably the diagnosis would just be "idiopathic ocular pigment production" or something like that.
