One of the more famous achievements of Avraham was his discovery in a single God. However there is no mention of this accomplishment anywhere in the Torah. In fact there is no conclusive proof that he was a monotheist anywhere in the simple text of the Torah. Why is Avraham's discovery and belief in a single God not recorded in the Torah itself?
Its strongly implied in Yehoshuah, in his final speech to the people (ch. 24):
'וַיֹּאמֶר יְהוֹשֻׁעַ אֶל-כָּל-הָעָם, כֹּה-אָמַר יְ'ה'וָה אֱלֹהֵי יִשְׂרָאֵל, בְּעֵבֶר הַנָּהָר יָשְׁבוּ אֲבוֹתֵיכֶם מֵעוֹלָם, תֶּרַח אֲבִי אַבְרָהָם וַאֲבִי נָחוֹר; וַיַּעַבְדוּ, אֱלֹהִים אֲחֵרִים. וָאֶקַּח אֶת-אֲבִיכֶם אֶת-אַבְרָהָם, מֵעֵבֶר הַנָּהָר, וָאוֹלֵךְ אֹתוֹ, בְּכָל-אֶרֶץ כְּנָעַן; וארב (וָאַרְבֶּה), אֶת-זַרְעוֹ, וָאֶתֶּן-לוֹ, אֶת-יִצְחָק.
While Terach served other gods, God took Avraham away from that, which implies that Avraham rejected polytheism and just served God. Yehoshuah then ends off with a choice to the people - do they return to the idolatry of their fathers across the river or do they serve God?
וְעַתָּה יְראוּ אֶת-יְ'ה'וָה, וְעִבְדוּ אֹתוֹ--בְּתָמִים וּבֶאֱמֶת; וְהָסִירוּ אֶת-אֱלֹהִים, אֲשֶׁר עָבְדוּ אֲבוֹתֵיכֶם בְּעֵבֶר הַנָּהָר וּבְמִצְרַיִם, וְעִבְדוּ, אֶת-יְ'ה'וָה. וְאִם רַע בְּעֵינֵיכֶם לַעֲבֹד אֶת-יְ'ה'וָה, בַּחֲרוּ לָכֶם הַיּוֹם אֶת-מִי תַעֲבֹדוּן--אִם אֶת-אֱלֹהִים אֲשֶׁר-עָבְדוּ אֲבוֹתֵיכֶם אֲשֶׁר בעבר (מֵעֵבֶר) הַנָּהָר, וְאִם אֶת-אֱלֹהֵי הָאֱמֹרִי אֲשֶׁר אַתֶּם יֹשְׁבִים בְּאַרְצָם; וְאָנֹכִי וּבֵיתִי, נַעֲבֹד אֶת-יְ'ה'וָה.
The people then affirm they choose 'ה since he is our God:
גַּם-אֲנַחְנוּ נַעֲבֹד אֶת-יְ'ה'וָה, כִּי-הוּא אֱלֹהֵינוּ...
First, it should be noted that although the questioner list "discovery of a single God" as an achievement of Avraham this is not the case. The Midrashic Literature does indeed teach that Avraham came to the conclusion that the universe has one single Ruler, but this was not a novel idea. Adam HaRishon clearly knew this as he was created and interacted with Hashem, so did his children and the entirety of mankind until the days of Enosh when "the name of Hashem began to be profaned" and mankind began to slowly progress toward the mistaken ideology of polytheism and idol worship (see Rambam Hilchos Avodah Zarah), but there were always individuals who maintained the tradition from Adam of the Unity and Oneness of God. The greatness that is ascribed to Avraham is not his discovery of a single God, which would imply that this concept was unknown prior, but rather that he came to this conclusion based on his own searching and striving for truth, whereas all those who came before him merely preserved the knowledge that they had received based on tradition. Avraham was one who strove for truth and therefore was unsatisfied until he had achieved this aim.
The questioner's second contention that there is no record of Avraham's monotheism in the Written Torah is also unfounded. Before bringing scriptural support we must first define the term "monotheism." Does monotheism simply mean not worshiping more than one deity? If so, there were many pagan and polytheistic cults throughout the ages which practiced this type of 'monotheism' - believing that there local deity was the only deity worthy of their sacrifices or affection, or (as a means of political domination) forcing others to adopt the worship of their patron deity. No, simply avoiding the worship of multiple deities is not monotheism. It is a matter of belief. Does the belief that one deity is more powerful than all the others qualify as monotheism? Again, no. There were many pagan cults which we would qualify as polytheistic which held this belief. So then, what does a monotheist believe? Only one deity? What does this mean? What is the definition of a deity? Is it simply a matter of semantics? Is calling the one power which one believes to be supreme "god" and calling all the others by another name considered monotheism? If one calls Zeus "god" and all the other deities of Olympus "angels" is that monotheism? No. Monotheism is the understanding that although there are a myriad of forces, physical and meta-physical, which exist in the world all of the forces have one Creator, one Source, and are sustained by only one Force. That everything in the world is dependent on one and only one Entity. This one Entity is God. This belief is clearly expressed in Genesis 14 by both Malki-Zedek (14:19-20) and Avraham (14:22)
וַיֹּ֥אמֶר אַבְרָ֖ם אֶל־מֶ֣לֶךְ סְדֹ֑ם הֲרִימֹ֨תִי יָדִ֤י אֶל־יְהוָה֙ אֵ֣ל עֶלְי֔וֹן קֹנֵ֖ה שָׁמַ֥יִם וָאָֽרֶץ׃
Avraham clearly states his belief in one Entity who is the Creator and Source of everything (Shamayim and Aretz)
Now, the question of why Avraham's theological positions and his journey to achieve them are not a focus of the Written Torah is indeed a great one. One which has been pondered by many scholars. I would like to humbly suggest that the rationale of the Written Torah in leaving out this information is because the Book of Genesis is not a book of theology but rather of righteousness. The focus of Genesis, in particular the accounts of the Avos, and in more particular the account of Avraham is about Derech Eretz following the maxim "Derech Eretz kadma la Torah." In fact, Avraham is praised as being one who teaches his household this message:
כִּ֣י יְדַעְתִּ֗יו לְמַעַן֩ אֲשֶׁ֨ר יְצַוֶּ֜ה אֶת־בָּנָ֤יו וְאֶת־בֵּיתוֹ֙ אַחֲרָ֔יו וְשָֽׁמְרוּ֙ דֶּ֣רֶךְ יְהוָ֔ה לַעֲשׂ֥וֹת צְדָקָ֖ה וּמִשְׁפָּ֑ט
One might argue that this is what Rashi is alluding to in his answer to the question of why the entire Book of Genesis is included at all, "So that the nations of the world will not accuse you of being robbers." The entire Book of Genesis is also called "Sefer HaYashar" meaning the "Book of the Upright" which is a reference to the Avos.
מאי ספר הישר א"ר חייא בר אבא א"ר יוחנן זה ספר אברהם יצחק ויעקב שנקראו ישרים Source
The monotheism of Avraham Avinu was not as opposed to the Christian idea of Trinity. It was as opposed to idol worship. That was the polytheism which existed. Worshiping God is a completely different paradigm from idol worship. The fact that he worshiped God is clear in many places.
His unique achievements are that he came to this conclusion on his own, far from Shem. But more importantly, he actively spread God's name. This we see in the Pasuk in Bereishis 12:5 that Avraham took along the souls that he made. Rashi quotes the Medrash that he taught them and converted them. In Mishlei 11:31 it is used the same way, as Rashi points out there. We also find this unique praise about Avraham that he will pass the faith down.
The Gemara in Brachos points out that Avraham was the first to call Hashem by the name of א-ד-נ-י. This shown us that he did indeed have a unique and new approach. He is the first to introduce Hashem into the world as its immediate, reachable, and personal Boss. This, as opposed to Malki Tzedek who referred to Hashem as the High and Mighty.
it does allude that he was thrown into a furnace at Ur Kasdim for his refusal to worship idols. as the midrash expounds the verse, "When Terah had lived seventy years, he begot Abraham, Nahor and Haran... Haran died in the presence of Terah his father, in his native land, in Ur Kasdim" (Genesis 11:26-29).
Furthermore, that God chose to grant him prophecy and chose in him and his descendants to perpetuate Judaism certainly implies he believed in monotheism which is the cornerstone of judaism
Furthermore, the talmud (yoma 28b) says that Avraham kept the whole torah as it says: "Because that Abraham hearkened to My voice kept My charge, My commandments, My statutes, and My laws" (Bereishis 26:5). And belief in monotheism is the cornerstone of Judaism as the introduction to shaar yichud of chovos halevavos explains. So certainly he must have kept that.