Gemara

R. Avahu explained to R. Yitzchak bar Yosef how we establish the Mishnah. R. Yitzchak learned this from him 40 times, until it was [solidly ingrained in him] like in his pocket.

Temurah 14b - Question: Rav Dimi wanted to send an answer to Rav Yosef in a letter. One may not write Oral Torah!

(R. Aba citing R. Yochanan): One who writes Halachos is like one who burns the Torah. (One may not save it from a fire on Shabbos. Alternatively, one must destroy it);

Answer: Perhaps one may write a new insight (e.g. Rav Dimi's answer).

Support: R. Yochanan and Reish Lakish used to look in a Sefer of Agadata (non-Halachic teachings) on Shabbos;

They permitted this due to "Es La'asos la'Shem Heferu Torasecha." It is better to uproot a law of Torah in order that Torah will not be forgotten.

Rishonim

Rambam (Hilchos Yom Tov 7:14): During the Mo'ed one may write calculations and calculate his expenditures, for one is not careful to write them nicely, so it is Ma'aseh Hedyot (non-professional).

Ra'avad: This is a need of the Mo'ed.

Question: May one write calculations of Tekufos (equinoxes and solstices) during Chol ha'Mo'ed, just like a Tosefta permits writing calculations? This is like Devar ha'Avud, lest he forget them. If one heard a Chidush from his Rebbi, or had an idea and fears lest he forget it, it should be permitted to write it.

Answer (Rashba 3:273): Presumably, even if the Tosefta is like you say, one may not write calculations of Tekufos. Calculations of expenditures, or new ideas in Halachah, are like Devar ha'Avud, lest he forget. The calculation of Tekufos is known to one who needs it and is expert in the calculation. This is not a new Chidush, rather, like writing something he received orally, or like copying a Sefer. I am unsure about the Tosefta, for Chachamim said that one may not correct even one letter, even from the Sefer Torah of the Azarah. Why aren't we concerned lest they forget [to correct it]? One should correct all Seforim, lest he forget! Also, it is a Mitzvah not to leave an errant Sefer in his house - "v'Al Tashken b'Ohalecha Avlah." One will write his explanations and Chidushim, make compositions and write, and make his Mo'ed Chol! He will write Tefilin, even for others, for it is a Mitzvah! Rather, it seems that we permit writing only calculations that pertain to others, which depend on memory. Perhaps he will forget and lose his money. A Chidush that one heard or thought of, depends on understanding of the heart. After the Mo'ed he is wont to review his teachings, and understand like he understood before. If he fears lest he forget, he can repeat it many times, so it will be fluent on his lips, like it says "he learned it from him 40 times, until it was like in his pocket." They did so before it was permitted to write the Mishnah and Gemara. One may not correct Seforim, for he is wont to return [after the Mo'ed], see in them and fix them.

Shitah Mekubetzes (14b:4): Some texts say "Rabanan rely on their Girsa (the text they know by heart). Since they can forget, they write and store it away, and when they forget, they look in a Sefer.

Poskim

Shulchan Aruch (OC 545:1): It is forbidden to write on Chol ha'Mo'ed, even to fix one letter in a Sefer.

Shulchan Aruch (7): Whatever one may write on Chol ha'Mo'ed is permitted without a Shinuy. What is forbidden is forbidden even with a Shinuy.

Shulchan Aruch (9): If one heard a Chidush, he may write it so he will not forget it. Similarly, if he saw a new Sefer, he may copy it if he will not be able to copy it after the Mo'ed.

Beis Yosef (DH v'Chasuv): Hagahos Semak permits writing Chidushim on Chol ha'Mo'ed. I support this from Temurah 14b. We asked from R. Aba, who said that one who writes Halachos is like one who burns the Torah. We answered that perhaps a new insight is different. I.e. he feared lest it be forgotten. This shows that even when it was forbidden to write oral Torah, a Chidush was permitted. We can permit also on Chol ha'Mo'ed for this reason.

Beis Yosef (DH umi'Mah): The Tur says that the Rosh permitted him on Chol ha'Mo'ed to copy an answer to a question. It seems that he holds like the Tosefta, unlike the Rashba. The Rambam permits writing calculations because they are Ma'aseh Hedyot. Perhaps he would similarly permit a Chidush, for one is not careful to write [the letters] so precisely, since he intends just so he will not forget it.

Beis Yosef (DH v'Gam): Also R. Yerucham permits writing a Chidush that he heard, lest he forget it, or to copy it, in case he will not be able to copy it after the Mo'ed. There is no greater Devar ha'Avud. Also R. Peretz says so, but he himself was stringent to write it with a Shinuy.

Taz (13): "They are new to him" connotes that we do not distinguish whether he heard it, or thought of it himself, if he is capable. The concern lest he forget applies to both of them. Often one cannot remember what he himself was Mechadesh. Seemingly, this Heter is only if he already heard, but one may not hear l'Chatchilah in order to write it. This is like making Devar ha'Avud. However, this is wrong. One is commanded at all times to be Mechadesh in Torah. I would add another reason to permit writing Chidushei Torah, even without concern for forgetting. At every moment one is obligated to toil in Torah and be Mechadesh according to his ability. One cannot say that he will wait until after Yom Tov and then write his Chidush, for at that time he has a different Chiyuv, i.e. to be Mechadesh other Chidushim. If he will delay writing what he learned during the Mo'ed, he will need to learn it again what he already learned and remind himself what he was Mechadesh. This will be Mevatel him from learning new Chidushim at that time! This is the greatest loss there is, to lose time. The Bach wrote his Perush on the Tur also during Chol ha'Mo'ed. It seems that one who writes Chidushim on Chol ha'Mo'ed must not write thick (Ashuris) letters, rather, Meshita (thin letters, somewhat similar to Rashi script). If he needs a big letter, he should write it broken.

Mishnah Berurah (48): If one needs a Sefer to learn from during the Mo'ed, he may fix and bind it himself, and all the more so through Nochrim, for it is Devar ha'Avud. However, one may not fix a parchment in order to write on it after the Mo'ed. The Acharonim say that whatever one may write, he may make a quill and ink. Eliyahu Rabah was unsure whether or not one may be Mesartet (scratch a line to enable writing straight) for the sake of things permitted to write. Leket ha'Kemach forbids, for Sirtut is only for the sake of beautifying the writing. It is permitted only for the sake of Tefilin and Mezuzos, for one may not write them without Sirtut.

Kaf ha'Chayim (79): Also the Magid told the Beis Yosef that one may write Chidushim on Chol ha'Mo'ed, for it is Devar ha'Avud. R. Yakov of Mirvish asked of Shamayim, and received this answer. Chidushim in Torah are the dearest treasure to Hash-m.