The Oxford English Dictionary notes that "By our Laken" or "Byrlaken" is an now obsolete english expression meaning "by our Lady". "Laken" or "Lakin" are derived from "Lady +
Kin".
The OED also quotes the following usage of the word:
More (Apol. iv. Wks. 849/2, 1533): "By our lakens brother husband... yet woulde I rather abyde the perill of breding wormes in my bely."
1610 Shakespeare (Temp. iii. iii. 1, 1610): "By'r lakin, I can goe no further, Sir, My old bones akes."
1616 [W. Haughton] (English-men for my Money C4, 16161): "Bir laken sirs, I thinks tis one a clocke."
Could the name Laken possibly be related to the above?