I'm a baal teshuva with Czech Jewish ancestry. I have an interest in the minhag of my ancestors as well as in common cultural practices, and perhaps pass them on where I can. Have they been documented anywhere?
This psalm (Song for the Sabbath Day) is usually recited toward the end of the Kabbalat Shabbat service. Traditionally, reciting this psalm was the point when the worshiper began to observe the restrictions of Shabbat. This created a conflict when under the influence of Arizal (based on Shabbat 119a), people would go out into the fields to greet the Shabbat Bride and Queen (L'cha Dodi). They would then say Mizmor Shir the first time to complete the Kabbalat Shabbat service (but not start observing the Shabbat yet), walk back into Prague to the shul, repeat Mizmor Shir again (really starting the Shabbat this time), then begin Ma'ariv.
Over time, they got away from the custom of going out into the fields, but the custom remained to say Mizmor Shir twice with a Kaddish in between them. This was seen as strange and remarked on by several observers from out of town in the nineteenth century.
The source of this information is:
Why Jews Do What They Do: The History of Jewish Customs Throughout the Cycle of the Jewish Year pp. 12–21 by Daniel Sperber (translated by Yaakov Elman)
I do not have it in front of me right now, so I apologize if I have made any mistakes in recalling this from memory. If anyone has any corrections, I welcome them.
According to the סדר עבודת ישראל, whilst מנהג פולין say אב הרחמים on שבת מברכים החודש only during ספירת העומר, it says that ק''ק פראג (Prague) also say it on שבת מברכים החודש מנחם אב. http://www.hebrewbooks.org/pdfpager.aspx?req=42807&st=&pgnum=252