No, I would say it's not very common. I would never say someone "took lodging" with their parents -- it suggests that the parents run a boarding house and the son is paying rent. " (2) isn't bad -- it somehow suggests (to me at least) that the birds considered several different places and selected this one as the most desirable.
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MrPernickety In passing, could you tell me if there's any significant difference between "take lodging" and "take up lodging" ?I agree with Khoff, except for the the birds' selectivity. It may well be the case, but I don't hear it as implied in your sentence.
AvangiI hear "take up lodging" as the beginning of an indefinite stay, possibly long term.Hi, Avangi
"Take lodging," except with birds and other critters, strikes me as very temporary:
"We took lodging for the night."