In chapter 13 of Tarzan of the Apes (His Own Kind), the notice posted by Tarzan reads:
THIS IS THE HOUSE OF TARZAN, THE KILLER OF BEASTS AND MANY BLACK MEN. DO NOT HARM THE THINGS WHICH ARE TARZAN'S. TARZAN WATCHES. TARZAN OF THE APES.
Of course he did not pronounce God as you or I would pronounce His name, for Tarzan knew naught of the spoken language of his English forbears; but he had a name of his own invention for each of the little bugs which constituted the alphabet. Unlike the apes he was not satisfied merely to have a mental picture of the things he knew, he must have a word descriptive of each. In reading he grasped a word in its entirety; but when he spoke the words he had learned from the books of his father, he pronounced each according to the names he had given the various little bugs which occurred in it, usually giving the gender prefix for each.
Thus it was an imposing word which Tarzan made of GOD. The masculine prefix of the apes is BU, the feminine MU; g Tarzan had named LA, o he pronounced TU, and d was MO. So the word God evolved itself into BULAMUTUMUMO, or, in English, he-g-she-o-she-d.
Similarly he had arrived at a strange and wonderful spelling of his own name. Tarzan is derived from the two ape words TAR and ZAN, meaning white skin. It was given him by his foster mother, Kala, the great she-ape. When Tarzan first put it into the written language of his own people he had not yet chanced upon either WHITE or SKIN in the dictionary; but in a primer he had seen the picture of a little white boy and so he wrote his name BUMUDE-MUTOMURO, or he-boy.
To follow Tarzan's strange system of spelling would be laborious as well as futile, and so we shall in the future, as we have in the past, adhere to the more familiar forms of our grammar school copybooks. It would tire you to remember that DO meant b, TU o, and RO y, and that to say he-boy you must prefix the ape masculine gender sound BU before the entire word and the feminine gender sound MU before each of the lower-case letters which go to make up boy—it would tire you and it would bring me to the nineteenth hole several strokes under par.
In-universe, is it ever explained how he came to use Tarzan as the spelling of his name? The paragraph above would indicate that he would have switched to using "White Skin" as his name once he learned those words, but instead, we see that he used Tarzan.
Oddly, Burroughs calls it a "strange system of spelling", whereas it should be a strange system of pronunciation. If he did assign sounds to the various characters, and using (g - LA), (o - TU) as representative examples, he could have wrote down "Tarzan" in about as many characters, using different alphabets of course.