The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), in Siman 300 (Se’if Katan 3), writes that it is customary after the recitation of Habdala on Mosa’eh Shabbat to recite special hymns and prayers. The Rama (Rabbi Moshe Isserles of Cracow, Poland, 1525-1572), records (in Siman 295) a more specific custom to mention Eliyahu Ha’nabi on Mosa’eh Shabbat, and pray that he should soon come to herald the Jewish people’s redemption. The Mishna Berura explains this custom based on the Gemara’s comment in Masechet Erubin that Eliyahu is not going to come on Ereb Shabbat or Ereb Yom Tob, when Jews are busy making their preparations for Shabbat or Yom Tob, and would have to interrupt these preparations to greet Eliyahu. Once Shabbat is over, Eliyahu is again able to come, and we therefore recite prayers asking that he should arrive to announce the final redemption.
Additionally, the Eliyahu Rabba (Rav Eliyahu Shapiro of Prague, 160-1712) records a custom to recite the words "Eliyahu Ha’nabi Zachur La’tob" 130 times on Mosa’eh Shabbat. Some explain that this custom is based upon the numerical value of the name "Eliyahu Ha’nabi," which is 120 (52 + 68), and when we add the ten letters of these two words, we arrive at 130.
The Abudarham (Spain, 14th century) explains that we mention Eliyahu Ha’nabi after Habdala because when he arrives, he will distinguish between the "Kesherim" and "Pesulim" – those who are permissible to marry into the Jewish nation, and those who are not. It is thus appropriate to pray for Eliyahu’s arrival on Mosa’eh Shabbat, after Habdala, which speaks of the distinction between Shabbat and the weekdays. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) explains the connection between Eliyahu and Mosa’eh Shabbat based on the tradition that the Jewish people would earn redemption if they properly observe Shabbat two consecutive weeks. Therefore, after Shabbat, we express our hopes that our nation has properly observed two Shabbatot and Eliyahu can thus now arrive to announce the redemption. Furthermore, the Midrash comments that every Mosa’eh Shabbat, Eliyahu Ha’nabi enters Gan Eden, sits underneath the Tree of Knowledge, and writes the merits of Am Yisrael. This is yet another reason why we speak of Eliyahu Ha’nabi on Mosa’eh Shabbat.
The Mishna Berura also cites the Eliyahu Rabba as recording a custom observed by the "Medakdekin" (those especially meticulous in Halachic observance) to recite additional songs and hymns on Mosa’eh Shabbat. He mentions in particular the hymn "Ribono Shel Olam Ha’hel Et," based on the Yerushalmi. There are also those who light candles on Mosa’eh Shabbat in dark areas, in commemoration of King Shaul, who rose to kingship in the merit of his grandfather, Ner ("candle"), who was so named because he would light torches in the dark roadways.
The Rama (295) records the custom instituted by the Arizal (Rav Yishak Luria of Safed, 1534-1572) to recite on Mosa’eh Shabbat "Ve’yiten Lecha," a collection of verses that speak of blessing. The Mishna Berura explains that reciting these Pesukim serves as an auspicious omen of blessing for the coming week. Some have the custom to recite "Ve’yiten Lecha" before Habdala, whereas others recite it after "Habdala." Our community’s practice is to recite it after Habdala.