Hojicha is green tea that has been roasted. The Ministry of Agriculture, Forestry and Fisheries describes it as tea made by roasting sencha, bancha and the like. That one sentence is the whole definition, and it is worth holding on to, because most of the confusion around hojicha comes from setting it next to names that describe something else entirely.
Hojicha names a process; stem tea names a part
The step of roasting tea is called baisen. The name hojicha points at that step and at nothing else. It tells you what was done to the leaf — not which part of the plant went in, and not where it was grown.
Stem tea, kukicha, is the opposite kind of name. It is tea gathered from the stems of gyokuro or sencha, so the name describes the part used. Because the two names are about different things, they do not divide into separate categories: hojicha made from stems exists, and is common. Stem tea (bōcha) may or may not have been roasted, and a pack labelled hojicha may or may not be stems. You have to read both halves of the description.
Genmaicha roasts the rice, not the tea
Genmaicha is sencha or bancha combined with roasted brown rice and similar grains. The roasting is real enough, but it happens to the grain. In hojicha, the tea itself goes over the heat. 'What gets roasted' is the easiest handle for keeping the two apart.
Two accounts of where hojicha began
Ask when hojicha began and more than one answer comes back. Two published accounts are worth reading — on the understanding that they are not answering the same question.
The tea merchant Yamamotoyama writes that, from the end of the Taisho era into the early Showa era, tea merchants in the Yamashiro area of Kyoto Prefecture, troubled by leaf that would not sell, devised hojicha in order to make use of stock piled up unsold. Time, place and trigger are all given. The company ends the sentence with 'it is said': it presents this as a story that has been handed down rather than as established fact, and shows no primary document. Read as an account a tea merchant is passing on, it holds; read as a settled date, it does not.
The second account is from Ishikawa Prefecture. The Ministry of Agriculture, Forestry and Fisheries, in its encyclopedia of traditional Japanese foods, writes that in Meiji 35 a Kanazawa tea merchant, Hayashiya Shinbei, developed a way to make use of the stem portion — a by-product of drying tea leaves for long storage. He roasted and fired the stems and sold the result as Kaga bōcha, and it became very popular for its distinctive aroma and flavour and for its everyday price. The year, the place and the person are all named.
What that page does not say is that hojicha across Japan began there. It is an account of one region's tea.
Why we leave the question open
The Kyoto account is about roasting leaf that had not sold. The Kaga account is about finding a use for stems. The raw material differs, the region differs, and the period written about differs. Setting the two against each other to decide which is the 'real' origin means reading past what each source actually says. If you want to compare them, you first have to line up the axes — and these two do not line up.
Neither source gives an independent record of when hojicha spread across the country in the form we drink now. We have not found one. So the honest answer is that there are several accounts, each with its own scope, and we keep them separate rather than merging them into a single story. If a document turns up, we will add it here.