Our son was born last week does not sound weird to me. Actually Our son was born in the last week sounds a little bit unusual. But thinking about it, I find the latter one more correct grammatically. It feels like this was the original usage but people omitted "in the" in some time in history. What is the opinion of native speakers on this?
P.S. A French learning website marked "Our son was born in the last week" wrong as the translation of "Notre fils est né la semaine dernière".
'In the last week' suggests that you don't know when exactly your son was born, but are sure it was no more than one week ago, which would, as you observe, be an odd thing for a new parent to say. 'Last week' is natural though. The French website is right. La semaine dernière means 'last week', and la dernière semaine means 'the last week'.
'In the last week' means 'during the seven days prior to today' and 'last week' means 'during the calendar week prior to the current one'. They have different meanings.
To elaborate on your intuitions about grammaticality, many time expression modifiers that situate an event on the calendar do not require a preposition like in or on:
Others do require a preposition, however:
Some other kinds of temporal expressions can occur without a preposition:
But for establishing a time span relative to the present in which something happens, a marker like in is generally required:
And compare the following, which establish a point in time relative to the present, rather than a time span:
