The word tichon תיכון is of biblical origin, but the precise biblical usage isn't found much today. It means "middle", from toch תוך (midst, interior), in the same way that chitzon חיצון - "external" derives from chutz חוץ (outside). In the Bible, it is mostly used in describing construction, as in Shemot 26:28 - וְהַבְּרִיחַ הַתִּיכֹן בְּתוֹךְ הַקְּרָשִׁים - "the middle bar in the midst of the boards".
However, today the word is found mostly in three Modern Hebrew phrases. Let's take a look at them:
a) Yam Ha-Tichon ים התיכון. This phrase refers to the Mediterranean Sea. It's a little frustrating that this is what the sea is called in Modern Hebrew, because Biblical Hebrew has no shortage of names for it: Yam HaGadol ים הגדול - "the great sea" (Bamidbar 34:6, etc.), Yam Pelishtim ים פלשתים - "sea of the Philistines" (Shemot 23:31), and Yam HaAcharon ים האחרון - "the Western sea (literally "rear" sea, as they oriented themselves to the east.)" (Devarim 11:24, etc.)
So where does Yam HaTichon - the "middle sea" come from? Not actually from "Mediterranean", which literally means "midland". Rather it is a translation of the German Mittelmeer, which means "middle sea".
b) Mizrach Ha-Tichon מזרח התיכון. This is a direct translation of the English "Middle East", which we all know refers to the countries of southwest Asia and northeast Africa. Except that it's not entirely true. As Joel Achenbach writes in his book Why Things Are:
Q: Why do we always hear about the Far East and the Middle East but never the Near East?
A: The Near East is the Middle East; there isn't a Near East anymore. We start in the Middle, then go to the Far.
For centuries the term Near East referred, sensibly enough, to everything from Morocco to the Persian Gulf. The Middle East extended from there to Southeast Asia. The Far East included the nations along the Pacific. When World War II broke out, Britain transferred its Middle East military command from India to Egypt, to be closer to the action. The new station kept the old name. Gradually almost everyone picked up the new British nomenclature.
This of course includes Hebrew, where mizrach hatichon is the name almost exclusively given to the region.
c) Beit Sefer Tichon בית ספר תיכון. This is the most confusing of the three - I'm still not sure I've tracked down the etymology fully. Today it certainly refers to "high school", but as you might have guessed, the literal translation means "middle school." At first glance, one might assume (and I've seen a number of websites who claim) that high school is placed in the middle of elementary school and university. However, the senior Hebrew linguist Yechezkel Kutscher wrote:
The German “Mittelschule” – “high school” was first translated literally bet sefer benayim and today bet sefer tikhon.
This translation is rather old - Ben Yehuda mentions it in his dictionary, and when searching historic Hebrew newspapers, I found mention of "beit sefer tichon" as far back as 1895, but no mention of Kutscher's earlier phrase - בית ספר ביניים beit sefer benayim.
What did mittelschule originally refer to? Apparently, it was an "intermediate school" for the "middle ranks" or "middle class", as described here:
Parallel to the Volksschule was the Mittelschule, intended for the middle classes.
Or also here:
The tripartite secondary school system, with the Gymnasium or Oberschule for the children of the educated class, the Mittelschule for the middle ranks, and Hauptschule (main school) for the ordinary workers
I'm not familiar enough with either the German educational system or the early Zionist / Israeli educational system to fully described the influence of the former on the latter, but clearly it existed. For example, two of Israel's oldest high schools - the Herzliya Hebrew Gymnasium and the Hebrew Reali School, took their names from the German Gymnasium and Realschule respectively. In addition, as the Safa Ivrit website points out, the unusual nicknames for the high school classes: the 12th graders are called shministim שמיניסטים - "eighters", 11th graders are in shviit שביעית - "seventh", etc., is based on the German system, where secondary education would begin in fifth grade for eight years. This was the case in Israel as well until 1968, when the junior high schools - chativat beinayim חטיבת ביניים - were established, leaving high schools with only three or four years.