Mentioning Eliyahu Hanavi:
It is accustomed to mention the name of Eliyahu Hanavi after Havdala on Motzei Shabbos as a prayer that he should come and proclaim the redemption.
Eliyahu Hanavi writes the merits of the Jewish people: 
During the time of Motzei Shabbos Eliyahu Hanavi sits in Gan Eden under the tree of life and writes the merits of the Jewish people. It is for this reason that we mention Eliyahu on Motzei Shabbos.
Different Customs in mentioning Eliyahu: 
Some have a custom to recite all the verses which mention Eliyahu [see next].
Others write one is to say the name Eliyahu a total of 130 times. Some recite Eliyahu Hanavi 40 times, Eliyahu Hatishbi 40 times, Eliyahu Hagiladi 40 times and then recite again each one three times, concluding with Eliyahu Hanavi for a total of 130 times. They conclude “Bimiheira Yavo Eileinu Im Moshiach Ben David”.
Segula to prevent forgetfulness and to bring a successful week:
The Tur mentions the following custom: Some have a custom to recite all the verses that mention Eliyahu and doing so is a Segula for ones memory and to have a successful week.
 The reason for why he is mentioned specifically on Motzei Shabbos is because Eliyahu could not come on Friday due to it being a disturbance for the Jewish people which are preparing for Shabbos. Likewise he could not come on Shabbos if we hold there is an Issur of Techumin above 10 Tefachim. Therefore after Shabbos we pray that he should immediately come, since Shabbos has passed and he is now able to come. [ibid] See Shaareiy Geula “Haemuna Vehatzefuya” 7
 Elya Raba 295/9 brought in Kaf Hachaim 295/19; 300/5
 Ketzos Hashulchan 100 footnote 3
 Elya Raba 295/7 brought in Ketzos Hashulchan 100 footnote 3; Kaf Hachaim 295/20