In the Cairo Genizah a fragment of a midrash was discovered that, among other things, lists the birthdays of the shevatim. Ezra Chwat from the National Library of Israel transcribed the midrash and added footnotes comparing other sources (can be found here):
Reuven - 14th to the 9th month - 14th of Kislev
Shimon - 21st to the 10th month (28th according to Yalkut Shimoni) - 21st/28th of Tevet
Levi - 1st to the 1st month (16th of Nissan according to Seder Hadorot) - 1st/16th of Nissan
Dan - 29th to the 6th month (9th of Elul according to Seder Hadorot) - 29th/9th of Elul
Yehudah - 15th to the 3rd month - 15th of Sivan
Naftali - 5th to the 7th month - 5th of Tishrei
Gad - 9th to the 8th month (10th of Cheshvan according to Seder Hadorot) - 9th/10th of Cheshvan
Yissachar - 14th to the 5th month (10th of Av according to Seder Hadorot) - 14th/10th of Av
Asher - 2nd of Shevat (unique among the other dates to being the only month listed by name and not number) (20th of Shevat according to Seder Hadorot and 22nd of Shevat according to Shalshelet Hakabbalah) - 2nd/20th/22nd of Shevat
Yosef - 21st to the 4th month (27th of Tammuz according to Shalshelet Hakabbalah) - 21st/27th of Tammuz
Zevulun - 7th to the 7th month - 7th of Tishrei
Binyamin - 11th to the 8th month (which is also the day Rachel died) - 11th of Cheshvan