Eishes Chayil is indeed a beautiful song. But why do we say it specifically Friday nights (before Kiddush)?
3 Answers
According to this website it is Because Shabbos it self is a queen http://www.askmoses.com/en/article/205,152557/Why-do-we-recite-the-Aishet-Chayil-A-Woman-of-Valor-on-Friday-Night.html
The simplest answer is to show gratitude to the woman of the house who lead the preparations for Shabbos. She is likely tired and over-worked and deserves our sincere thanks!
Rabbi Eliyahu Lopian zt”l (1872-1970) recounted a great mystery which he set out to solve. When he first came to learn under the Alter of Kelm, Rabbi Simcha Zissel Ziv Broida zt”l (1824-98), he was in awe of the Alter’s wife. Rebbetzin Chaya Leah appeared to be a superwoman! She cooked, cleaned, cared for her three children and at the same time was dedicated to every need of the Yeshiva as well. How did she do so much and maintain her enthusiasm and vigor?!
This enigma was solved the first time that young Eliyahu was invited for a Shabbos meal. When they entered the house after Shul on Friday night, the Alter gave his wife a magnificently warm greeting and enumerated to her all of the appreciation he had for her hard work to organize, cook and care for the family’s needs. She was beaming. After making Kiddish and HaMotzie, the Alter smiled at the Rebbetzin and told her that her Challah was the sweetest and most delicious that he had ever tasted. The pride and joy found on the Rebbetzin’s face explained everything to Rabbi Lopian. When someone feels truly appreciated, recognized and validated, they experience a magical surge of energy that inspires them to continue their great accomplishments. That joy and fulfillment is truly powerful.
See the Otzer Dinim U'Minhagim quoted in this answer, discussing Shir HaShirim.
Quoting Minhagei Yeshurun, one of the answers is that Friday night is the time of intimacy between a man and wife.