This question is going to be very difficult to answer for three reasons. Firstly, because there is no such thing as a universal "Stone Age", secondly because the very idea of being transgender is a modern and culturally specific construction, and thirdly because analogous concepts in past cultures will vary greatly depending on the culture and so we can't generalize for every culture in a time period.

In regards to the "Stone Age" not really existing, different places on the planet experienced a "Stone Age" for different lengths of time and beginning and ending at different times. Additionally, "Stone Age" as a way of classifying time isn't really used by most archaeologists because it isn't specific enough. Very generally, most places on the planet will be divided into a "Paleolithic" ("Old Stone" age) and a "Neolithic" ("New Stone" age). Very generally, the Paleolithic in an area is characterized by stone tool technology and hunting and gathering subsistence or mixed hunting-gathering and horticulture. In contrast, the Neolithic in an area is generally characterized by the continued use of stone tools, but also the introduction of groundstone tools (so grinding stones, generally) and a change towards intensified use of wild cereals and the beginnings of plant and animal domestication. The origins of sedentary agricultural life are generally in the Neolithic.

That all said, these distinction are only useful in areas were sedentary agriculture became the primary subsistence strategy. In places were agriculture didn't supplant existing subsistence strategies, or areas which switched to nomadic pastoralism (i.e. herding), it isn't really appropriate to talk about a "Stone Age" at all.

For instance, in North America, the Paleolithic (what we call the Paleoindian era) generally ends around 2000-1000BC with the introduction of maize to the Southwest, and later to the Southeast. The Neolithic (what we call the Archaic period) generally ends around AD500 in the U.S. Southwest, and at a different time entirely in the U.S. Southeast. Here we mark the end of the Neolithic not by the introduction of metal tools (which is what marks the end of the Neolithic in Europe), but by the full adoption of sedentary agriculture. In contrast, in Europe, the Neolithic ends around 3000BC (but this varies depending on where in Europe) and involves the introduction of metal tools.

TL;DR: "Stone Age" isn't a specific era of history and is regionally specific. Additionally, it isn't appropriate to even talk about a "Stone Age" for certain areas and histories (like the Great Plains or parts of Central Asia) because a "Stone Age" doesn't describe the history of those areas very well.

As for the second point, the idea of being transgender is a modern, Western concept to describe a person experiencing gender dysphoria (the feeling that a culturally assigned gender and biological sex do not match an internal conception of self and the consequent distress). While there were probably people in the past who experienced gender dysphoria they would not have understood themselves to be transgender as that is a term and concept specific to the recent past and mostly to Western cultures. That said, there are certainly very many well documented systems of non-binary gender in a variety of societies, including the Berdache of many Great Plains tribes such as the Crow and Sioux. These often operate with an individual taking on the culturally specific qualities of the gender opposite of what they were assigned at birth based on their biological characteristics. All that said, it would be a mistake to call the people within these systems of non-binary gender "transgender" because that is a culturally specific term which doesn't capture all the nuance and specificity of these systems in other cultures.

Finally, even if we could talk about a single "Stone Age" and if the idea of being transgender was transferable to people in the past, treatment of these people would very much be dependent on the particular culture in which they lived. Despite all being nominally "Neolithic" cultures, treatment of certain classes of people will be very different in Neolithic China compared to Neolithic Europe or the Archaic Southeastern North America. The treatment of people not belonging to a binary gender system is entirely culturally dependent and not at all dependent on the subsistence practices and technologies of that society. There are documented non-binary genders among almost every kind of society, from traditionally hunter-gatherer groups like the aforementioned Great Plains tribes to sedentary agriculturalists like in Southeast Asia.

If you can narrow your question down a bit more you might be able to receive a better answer, but until then you might be interested in this AMA about the history of sexuality (less related to gender issues, but there is some information) as well as this Tuesday Trivia feature about alternate gender expressions.

this may be worth x-posting to /r/AskAnthropology since the Stone Age is generally pre-history