When evaluating the significance of the sun in Camus' The Stranger, we might focus on the connection between the day of the funeral (when Mersault's mother is buried) and the day that Mersault commits the murder.
On each of these days the sun is mentioned as being very hot, oppressive, and even painful.
We might conjecture that when Mersault encounters the extreme heat/sun on the beach he is (consciously or unconsciously) reminded of his mother's death and burial. The latter sections of the novel offer ample suggestion that Mersault is affected by his experiences (like his mother's death and his affection for Marie) even though he denies that these experiences are influencing him.
The espoused emotional and intellectual neutrality of Mersault's narration of his own psyche is brought into question by contrasting examples of deeply felt emotion and keenly observed, very human turmoil.
The second part of the novel deals with Mersault's denial of an inner life and the ramifications of this denial are indirectly examined in this section. However indirect the treatment of this idea may be, the notion that Mersault's actions are connected to an emotional inner-life is repeatedly suggested by almost all the characters that Mersault encounters in Part 2. Thus, while Mersault maintains an inner-dialogue that consistently denies emotional causes for his behavior, the novel creates a significant sense of doubt as to Mersault's ability to understand himself.
His behavior may be, ultimately, outside of his own control. His impulses may be inspired by emotional connections (such as the sun-mother's-death connection) that he either does not see, does not understand, or is unwilling to consider. The justice system's attempts to apply a strict morality to a person like this (who does not recognize his own emotional/psychological state of being) begins to illuminate Camus' notion of the absurd at work in The Stranger.
In any event, the sun is powerfully present when Mersault's mother is buried as well as on the day of the murder. The meaning of this connection and this symbolism is, perhaps, open to interpretation. The coincidence cited here may be intended to be understood as just that - a meaningless coincidence.