A number of years ago, David Bogner of Treppenwitz asked the same question. In the end he was less concerned about the etymology than he was about how much he enjoyed manning the grill for the soldiers in the area. Some of his readers suggested that the origin was Turkish. Another veteran blogger, Allison Kaplan Sommer, agreed, and wrote:
I made my little etymological discovery this year on my vacation in Istanbul, Turkey. I was touring one of the palaces, and they were showing us the little portable stoves that would be moved from room to room so that the sultans and their families could enjoy fresh tea anywhere. And the stoves were called....Mangals! So I guess the term is leftover from the Ottoman Empire days.
The best English translation for mangal is brazier, as defined here:
1. A metal pan for holding burning coals or charcoal.
2. A cooking device consisting of a charcoal or electric heating source over which food is grilled.
In Ottoman Turkey the mangal was primarily used for heat, and cooking over it was a secondary function. I found a number of sources from the 19th century describing the mangal (see here, here and here). The last one, from 1868, describes the mangal as follows:
The "mangal" is a large dish of live charcoal, and on the introduction of this into the room or ward they are dependent for warmth.Other regions under Ottoman rule also borrowed the word, and so we find in Romanian, Albanian and other regional languages that mangal means "charcoal". In modern Hebrew, mangal came to mean almost exclusively a stove for grilling, and now refers also to the barbecuing event itself.
However, where did the Turks get this word? According to an article by Amnon Shappira of the Hebrew Language Academy (Leshonenu Le'Am 45:3), the Turks borrowed it from the Arabic word mankal (מנקל) - also meaning stoves. Where did the switch from mankal to mangal occur? Shapira brings up the theory that it could be from Bedouins or even Yemenite Jews, who substitute "g" for "k". However, he writes that the use of the mangal was not as common on the Arabian peninsula, but more likely entered Turkish via Persian, which makes a similar consonant switch.
What does mankal mean in Arabic? The root of the word is נקל, which means "transport, transmit, convey". So as Allison wrote, the "little portable stove" became known as mankal. (The Hebrew acronym for CEO - מנכ"ל, is not related).
This Arabic root is the source of the word nagla נגלה - which originally meant "load" (as in donkey-load) and took on the meaning "round" or "trip" in Modern Israeli slang. In this case, according to Stahl, the switch from nakla to nagla is via Bedouin pronunciation.
Another word deriving from the same root is the game mancala (also mankala / manqala). This game (popular with my kids, although I haven't actually played it yet), involves transferring stones around a board.
There is actually one classical Hebrew word that might be cognate to the Arabic. That is the word makel מקל - meaning "staff" or "walking stick", and it is found in the Tanach (although far less frequently than the words mateh מטה and shevet שבט, of similar meaning.) The etymology of the word is unclear, and Ben-Yehuda lists many theories. One of them is that it is related to נקל. Stahl says this because the stick was used to move animals from place to place, as in Bamidbar 22:27:
וַיַּךְ אֶת-הָאָתוֹן בַּמַּקֵּל "and he struck the donkey with his stick"
I think perhaps another verse that could show a connection would be Bereshit 32:11, where Yaakov says "with my staff I passed over this Jordan" - כִּי בְמַקְלִי, עָבַרְתִּי אֶת-הַיַּרְדֵּן הַזֶּה
However, in the end, both Klein and Even-Shoshan say that makel might actually derive from the Egyptian word ma-qi-ra.
Let's end on a slightly more patriotic note. The official Hebrew word for mangal is matzleh מצלה, and nagla is masov מסוב. I haven't heard one yet for mancala...