Who is the author or what is the origin of the song commonly found in the haggadah called Echad Mi Yodeya?
The Encyclopaedia Judaica, cited in the Wikipedia article linked in the comments above, states as follows:
Contrary to the EJ's assessment that the Jewish song was preceded by the Christian one and merely derivative, Professors Philip Bohlman and Otto Holzapfel contend in their book Anthology: The Folk Songs of Ashkenaz (pp. 68-74) that the original provenance of the song is less certain:
R' Yedidya Weil, in his Marbeh L'sapeir commentary on the hagada (1791), mentions a report (43a, 47a) that this song and Chad Gadya can both be traced to a manuscript found hidden inside a wall in the study hall of the Rokeach (12th - 13th century) in Germany. This article says that the commentary reports that the manuscript was dated to 1406, but after skimming the commentary, I only found two paragraphs discussing this topic, and I didn't find such a date mentioned in either. Possibly unrelatedly, R' Weil does cite a parchment manuscript of a siddur from 1406 in his discussion on הא לחמא עניא (p. 4b), in his discussion on נשמת כל חי (p. 35a), and in his discussion on ישתבח (p. 37a).
The following is from the On the Main Line blog:
1 The first stanza of this folk song roughly translates to: "Good friend, I ask you!/ Good friend, what do you ask me?/ Tell me what is oneness?/ Oneness and unity is the Lord God who lives there/ and hovers there/ in heaven and on earth". Similar variants also exist.
2 In chapter 36 of הגדה שלמה, R' Kasher cites Nehemiah Brüll's Jahrbücher für Jüdische Geschichte und Litterature, IV, 1879, p.97, which, in a footnote on a biographical digression about R' Weil and his writings, mentions both the siddur from 1406 and the recovered manuscript from the beis midrash of the Rokeach. R' Kasher appears to connect the two, saying that the former is the source for the latter: " בסידור כ"י משנת ה"א קס"ו (1406) נאמר כי נמצא כתוב על קלף בבית מדרשו של ר' אלעזר רוקח בגרמיזא". Perhaps I missed something in מרבה לספר (or in Jahrbücher), but, from what I could tell, R' Weil simply attributed the story about the recovered manuscript from the Rokeach's beis midrash to a report that he heard ("אמרו שמצאו גנוז" and "שמעתי שמצאו זה... גנוז").
R' Kasher does not discuss the issue further, but he does echo the EJ's comment (above) regarding differing liturgical practices among Jews ("ובקהילות קוגין וסינגלי נהגו לזמרו בנוסח שונה בסעודת השבת שבשבעת ימי המשתה"), citing the "Goldschmidt Hagadah, p. 76".
The following is a footnote from an as yet unpublished Haggadah Yalkut (compiled by me). It discusses a possible 'origin' , or at least influence, on Echad Mi Yodea.
Shmuel bet 24:1-14 tells us that the mageifa, a plague sent in the aftermath of David’s ill-advised census, should have lasted for 72 hours. Meam Loez, based on Pesikta Rabbati, informs us that in response to David’s contrition, haShem reduced the plague’s duration to 36 hours and then announced He would consider further pleas for leniency. Immediately, sangerim tovim/effective advocates, come forward: mila, Shabbat, the Avot, the Aseret haDibrot, the luchot, and Torah (whose numeric values are, respectively, 8+7+3+10+2+5 = 35), and the plague was reduced to a single hour. (Another version of the midrash replaces the Aseret haDibrot and the luchot with the twelve shvatim.) I find it curious that here we have most of the numbers/identities that appear in Echad Mi Yodea, and that they are described as sangerim tovim shel Yisrael. Could those who composed this song have had this midrash in mind? (Consider, too, the fact that Divrei haYamim Alef 21:16, the prooftext cited in Maggid linking zro’a netuyah to a sword, is taken from that sefer’s retelling of this census episode and describes the sword-bearing angel as a mashchit/destroyer, the same term used in Shmot 12:23 at makkat b’chorot.) So far, I have found only the Safrai haggadah and David Arnow’s Making Lively Seders sharing ‘my’discovery, with only the latter also musing on this as a source for Echad Mi Yodea.
Related to this is another footnote:
There is a curious comment in the Haggadah Tzuf Amarim (by R’ Moshe Kleinman). He cites Pirkei D'Rabbi Eliezer (28) which states that in the merit of nine, we were redeemed from Mitzrayim. These nine were comprised of the three avot, the four imahot, and the two sources of blood (circumcision and the korban Pesach) shed before the Exodus. Parallel to this, he writes, women spent their nine months of pregnancy in fear and trepidation lest their child be male and, if so, seized and drowned in the Nile. While this pair of nines is not a completely convincing rationale for the inclusion here of ‘nine months of pregnancy’, it has the advantage of a structural affinity with a midrash [cited above].