Shabbos Shuva, the name for the Shabbos between Rosh HaShana and Yom Kippur, most directly gets its name from the first verse of the dramatic and poignant Haftora read on this Shabbos: "Shuva Yisrael 'ad Hashem Elokecha ki chashalta Ba'Avonecha" - "Return O Israel to Hashem, your G‑d, for your have stumbled with your sin."
Many contemporary writers have noted additional connections between the themes of Shabbos and repentance. Among these explanations is a fascinating exposition presented by the late Slonimer Rebbe zt"l, Rav Shalom Noach Berzovsky, in his Nesivos Shalom. The language of the Torah concerning the first Sabbath of Creation, "And Hashem finished His work on the seventh day" implies that something was created on Shabbos. This seemingly contradicts other verses indicating that the work of Creation was completed in six days. Expanding on statements by Chazal, Rav Berzovsky explains that Shabbos was the first day of re-creation. Nothing in the world exists fundamentally. It is only through the continued Will of G‑d that all of the billions of molecules in the universe and all its myriad laws continue. In the language of the prayers, "M'chadeish b'tuvo b'chol yom tamid ma'asei b'reishis" - "He renews with His kindness each day, constantly, the acts of creation."[1] This phenomenon of re-creation rooted in the very Creation, first occurring on the first Shabbos, serves as the cornerstone of the ability to change. Through the teshuva process, the individual breaks with his past and becomes a new being. R. Moshe Chaim Luzzato and others point to this aspect of repentance as that which allows for the erasure of the past evils acts. They are not attributed to the person who sinned because he is now someone else.
Rav Aharon Kahn shlita, suggested a similar approach in his analysis of a midrashic statement on a passage oft-recited on Rosh HaShana. "Tik'u bachodesh shofar, bakesseh l'yom chageinu" - "Blow the shofar at the time of the renewal of the month…." Yalkut Shomoni, expanding on the verb-forms of the words "baChoDeSH" and "SHoFaR", comments: "CHaDSHu ma'aseichem, SHaPRu ma'aseichem" - "Renew your actions, better your actions." Sometimes one may merely "fine-tune" one's past actions to achieve perfection. Quite often though, one needs an entirely different outlook, one which is G‑d-centered and not self-centered, in order to properly repent. This is the mystery of the human being's ability to take advantage of the G‑d-granted ability to perform true teshuva: to re-create one's life and devote it to the service of HaKadosh Baruch Hu just as the Creator Himself constantly re-creates His world.
[1] There is an interesting physical, natural phenomenon highlighting this very concept - that of constant cell division, the emergence of new cells and the death of the old while the organism remains the same.