There was a conversation between Joonas Ilmavirta and I in CONLOQVIVM, during which we attempted to figure out what the appropriate translation for the phrase "high school" (specifically of the American variant) would be in Latin. Needless to say, we did not come to a conclusion.
I am aware that the Roman concept of school was very different from what we would think of today, and as such, there will likely be no pragmatic translation (or even a mostly literal one).
TL;DR: what would be the best way to translate the phrase "[american] high school"?
If you want a single word meaning "high school" specifically, I think the closest would be lycēum. I think the word schŏla "school" would also be appropriate in reference to a high school.
As you note, the concept of a "high school" seems to be modern, so there is probably no exact classical equivalent. Joonas mentioned the word lycēum in chat, and I think this would be a good choice because many modern European languages use a word based on this to refer to a secondary school. (According to Wikipedia, this usage originates from Napoleon's educational reforms in the early 19th century.) The online Latin dictionary Latdict has the following entry:
declension: 2nd declension
- gymnasium of Aristotle
- high school (Ecc)
[...] Source: “Oxford Latin Dictionary”, 1982 (OLD)
In many contexts that I can think of, it would not be necessary to indicate that one is talking about a high school specifically rather than some other kind of school. In cases like that, I think the word schŏla, -ae (f.) would work—it is the source of the English word "school", and can have the same meaning in Latin. The "Brooklyn Latin School" (an American high school) seems to translate its name into Latin as "Schola Latina Brookliniensis"; the History page mentions the older Boston Latin School which seems to go by the name "Schola Latina Bostoniensis" in Latin.
My impression is that there is another Latin translation for "school" that is used particularly for primary schools—lūdus, -i (m.)—so in practice schola may not even be as vague as English "school". I'm actually not sure about the exact differences in meaning and usage between schŏla and lūdus, though: it seems probable that they have some overlap, as I found them both translated as "the School" in "A First Latin Dictionary" by William Smith (1862).
First of all, I think @sumelic's answer is excellent (and I've upvoted it), but I think it would be remiss not to mention a reasonable alternative, gymnasium.
In specifically American contexts, it's specifically standard to use 'schola': Schola Latina Bostoniensis (Boston Latin School, 7–12); Schola Latina Puellis Bostoniensis (Girls' Latin School, 7–12); Schola Daltoniana (Dalton School, K–12); Schola Sanctæ Mariæ (St Mary's School, 5–12); Schola Præparatoria Georgiopolitana (Georgetown Visitation Preparatory School, 9–12); Schola Latina Roxburiensis (Roxbury Latin School, 7–12); Schola Sancti Pauli (St. Paul's School, 9–12); Schola Latina Brookliniensis (Brooklyn Latin School, 9–12); Covingtonensis Latina Schola (Covington Latin School, 7–12); Schola Amicorum Abingtonensium (Abington Friends School, 9–12); &c.
Because of the way the US uses 'college' to refer to the first four years of university, it's less common to use the more proper 'collegium' the way many British schools do. Still, you see it occasionally, apparently most often with Catholic schools: Collegium Aquinatis Sancti Bernardini (Aquinas HS, 9–12); Collegium Sancti Caroli Borromeo (St Charles Preparatory School, 9–12); Collegium Sancti Benedicti (St Benedict's Preparatory School, K–12).
Private institutions are also often partial to 'academia' (after Plato's old haunts): Academia Phillipiana (Andover, 9–12); Phillip. Exoniensis Academia (Phillips Exeter, 9–12); Academia Sancti Xaverii Cincinnatensis (St Xavier HS, 9–12).
Newer American schools who want to distinguish themselves from ludi litterarii or scholae grammaticales now sometimes use 'schola superior': Schola Superior Archiepiscopi Moeller (Moeller High School, 9–12); Schola Superior Universitaria (University HS, 9–12). You could follow the Latin Wiki and take that as a calque for 'high school' but it's worth noting that in Europe, the same term is more often used for colleges and universities.
Similarly, the Latin wiki will sometimes calque the modern terms as 'educatio' or 'schola secundaria', although in classical Latin the adjective would've carried a pejorative 'second-rate' association.
'Lycæum' (after Aristotle's old haunts) is common in France but meant something else entirely in the US and isn't used in HS latinizations. The places where it has been used in English usually refer to vocational schools, not proper liberal-arts high schools; the places where has been used for actual high schools tend to be emphasizing their Frenchness.
To complement the other answers suggesting good translations, let me point out that one should not translate "high school" literally. One could form something like schola alta or schola superior (or with ludus instead of schola), but that would probably not be understood well.
The German, Swedish, and Finnish literal translations of "high school" or schola alta ('Hochschule', 'högskola', 'korkeakoulu') mean a university of some kind. In addition, the Finnish 'yläkoulu' ("upper school") is for children between roughly 13 and 16 years of age, and the same literal translation would fit the French 'école supérieure', a kind of university. These two would correspond to schola superior.
While direct translation is always possible, one should be aware of its perils. The plain schola is quite vague, and gymnasium can also be understood as a "gym". It seems to me that by far the safest Latin translation of "high school" as a general term is lyceum — other options are easier to misinterpret.
However, terminology is much freer when you translate the name of an individual school. Then you are free to use just about any word meaning "school", unless there is an important disambiguation to make.