There is a common minhag (custom) to sing Mizmor l'David at Seudah Shlishis, the third Sabbath meal. It is common to sing this song three times in a row and I was asked this Shabbos by a friend why this is so. None of my obvious points of references mentioned the reason, does anyone have a source and/or explanation (preferably "and")?
The source for my answer which follows below is from a Hebrew Book. The translation and shortening of the article is mine.
זמירון המקדש –הרב ישראל אריאל
All credit to Rabbi Ariel who states:
This Psalm, taken from Tehilim, summarizes by employing song and metaphor, the long journey of the Nation of Israel from the time of the exodus from Egypt, through the long exile of the Jewish people. The poet of the Psalm utilizes this description to portray the anticipation for the quick rebuilding of the Beit Hamikdash, (the 3rd Temple). The poet compares the Holy One to a Shepard, and Israel is presented as His flock. The period of desert travel (following the exodus) was a long tranquil period in the life of the Nation of Israel. During this era, Israel received everything already prepared; manna– from the heavens, and water – from the spring of bitter waters, and also, the pillar of cloud by day and the pillar of fire by night served as protection from above. In the Psalm, this period is compared to green pastures, as the Shepard provides his flock with abundant food. He makes the flock lie down in tranquility in pastures of grass. And He leads them to water in places of peace and rest. According to the Midrash on 23:5, You spread a table before me is a table that is filled with manna and quail.
According to the Midrash, there is no craftsmanship that is as shameful as that of shepherding, as the shepherd spends his life trudging along carrying his staff and pack. Notwithstanding this depiction of a shepherd, David refers to the Holy One as Shepherd. How can that be? David relied on Yaakov who called G-d Shepherd as stated in Bereishit 48:15 … The G-d who has been my shepherd from my birth to this day, so too David refers to G-d as Shepherd. – The Lord is my Shepherd I lack nothing. The “shepherd” is the most pleasant metaphor for a true leader, who worries about his flock, for the lone kid, and for the entire flock. Who is concerned that there is sufficient food and water for the flock and seeks to protect them from predatory animals.
In like manner, the period of exile is referred to a Valley of Darkness or Death, a period of testing for the Nation of Israel. How deep is their power of belief in the Holy One? And the poet attests to a strong belief as it is stated, though I walk through a valley of deepest darkness, I fear no harm, You are with me. And yes, occasionally the shepherd will strike his herd with his staff, and as the Midrash states that Israel refers to its tribulations during its exile as for the sake of the holiness of Your Name.
The poet ends his psalm by recognizing that all the trials and tribulations are aimed for the good., that the exilic period is a period of teaching and forging the nation. And in the end, I shall dwell in the house of the Lord for many long years. And that House/Temple will be the third Beit Hamikdash which will exist for a myriad of time.
It seems to me, that the metaphors of Table implies eating and that is what we do at the Seuda Shilishit.
The 3 times is Kabbalistic I am sure and I hope to return with a source.
May we all be blessed as one holy Jewish nation to celebrate together in a rebuilt Beit Hamikdash, in our time, in its proper place on Har Habayit.
Emunas Itecha 2 - Parshas Balak - Rabbi Moshe Wolfson Shlita says that similar to Succos where every day we go around the Bima once and on Hoshana Raba seven times on Shabbos we say Mizmor L'David once by each Seuda and three times by Shalosh Seudos, as Shalosh Seudos is Kollel all the Seudos.