What is the significance of The Stranger's lack of grief over his mother's death …?
L’Étranger is one book that definitely needs to be read in the original language.
It loses so much in translation, leaving it, as a comment in one of the other answers said, "boring, stupid, and a little sleazy".
In fact, the very first line, which in French establishes the character's view of the world, loses it completely in English, messing it up in at least two different ways:
Mother died today.
It makes "Mother" the significant part of the sentence, making us think that what follows will be about her. The formal and impersonal word "Mother" detaches Meursault from her.
But in the original French:
Aujourd’hui, maman est morte.
the idea is quite different.
It isn't a formal "Mother", it is "Maman", a much more personal and intimate term.
And the word order is different.
It leads with "Today", not "Mother".
What is most significant to Meursault is that it is today.
His character lives in the present, not dwelling on the past, not dreaming of the future.
He accepts what happens as it happens.
"Mother died today." is formal. It tells us about the woman and nothing about the narrator other than that he refers to her in an impersonal way. Her death was something she did, something that has already happened and already been consigned to the past.
A better translation, "Today, my Mom is dead." is informal and personal. Meursault is telling us about now, and we learn that he considers time important (something reinforced shortly later when he says that perhaps it was yesterday). Her death is part of his present, and he has affection for her.
To return to the original question though, it really isn't obvious that the narrator is lacking grief. He is simply a person that readily accepts the world he finds himself in, and a person without intimate social relationships and extravagant displays of emotion.