Q. What is the source that the Jewish people were redeemed in the merit of not changing their clothing?

A. The source seems to be Midrashic. The following sources are widely quoted as the origin for this well-known aphorism; Shemos Rabba 1:28, Vaykra Rabba 32, Shir Hashirim 4: 12, Psikta Beshalach 6: 6, Shochar Tov 114, etc. However a check on them will reveal that although they quote the preserving of names and language as factors for the redemption, the garments are not mentioned at all. A perush on the above-mentioned Pesikta (published by the Mekitzei Nirdomim Society, P. 219 n. 66) indeed vehemently claims that; “this was never said by our sages”.

Horav Shlomo Miller Shlit’a advised to consult the Torah Shleima Chumash and indeed in the Miluim on Shemos 3, he also wonders about the source for this widespread adage, and mentions that it is to be found on Midrash Lekach Tov, which is also known as Pesikta Zutressa (Devarim 26: 5.) A check there reveals that although it does mention that their garments were different from the Egyptians, nothing is said about that being a factor in the Geula. Torah Shleima on Vaera (Miluim 2) mentions different versions of the text of the midrashim that may be the elusive source. (See Veani BeHashem Atzapeh p. 207 – 208)
On the same Pesikta and from the Pesach Hagada on the above posuk (Devarim 26: 5) He went down to Egypt and sojourned there with a small number of people, and there, he became a great, mighty, and numerous nation; this teaches that they were distinguished (metzuyanim) there. Meforshim explain that they maintained their distinguished apparel. (Torah Shlemah ibid, Vayechi Yosef p. 27, Birchas Yosef 12 et. al.)

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a