WARNING: Spoilers for The Third Day episode 3, "Sunday - The Ghost"
What's the true meaning of the crickets in The Third Day? There's a Man vs. Nature element to the HBO miniseries, however, the subtext is rooted in psychological themes. In the third and final episode featuring Jude's Law's Sam, the crickets are directly referenced and correlated with two different concepts.
The four elements of matter are prevalent throughout The Third Day. Early on, Sam connects with the earth by visiting a forest location where his son was murdered. Specifically, he drops the boy's clothes into a stream of water, which is a symbolic act of letting go. After helping a young girl named Epona, Sam brings her home to Osea Island. There, he takes in the air and energy of the mysterious community, and eventually has nightmares involving fire. Sam acknowledges that he's been to Osea before, and his magnetic attraction to the island suggests that he's always been part of it somehow. The mysteries of The Third Day fully unravel in episode 3, "Sunday – The Ghost", in which crickets are a recurring visual motif. Sam attempts to escape imminent danger, but he's pulled back by a sense of familiarity and responsibility. As the crickets chirp, Sam feels "the darkness" looming over each and every decision.
Crickets show up immediately in episode 3. First, the insects appears on Larry's face while he and a preacher discuss Sam being the "true Father". Soon after, Sam learns the truth about his lineage from Professor Mimir (Börje Lundberg), who reveals that Law's character is actually the rightful Father of Osea. After Sam escapes from Larry in The Third Day, crickets once again show up all over the shore. When Jess (Katherine Waterston) saves Sam from drowning in the causeway, the two have an interesting conversation about the crickets. Sam reveals that his son went through a "cruel phase" by torturing insects; Jess states that crickets are a sign of rain, flooding, and even death.
From a psychological perspective, the insects in The Third Day are symbolic of Sam's overwhelming experiences - he's flooded with information, and framed as someone who's in touch with "the darkness." When Sam attempts to literally swim away from Osea, he's essentially trapped in the rolling waves of his own mind. As the pressure increases, Sam's fractured mind continuously affects his perception of reality. Numerous scenes can indeed be interpreted literally, certainly those in which Sam has conversations with other characters, but many sequences appear to be symbolic of Sam's frame of mind.
Midway through The Third Day episode 3 on HBO, Sam squashes a cricket with his foot, but fails to stop the proverbial rain. First, he discovers that Epona sacrificed her life to make up for Sam's failure to accept his fate as Osea's Father. Sam then learns that his son wasn't actually murdered, but was rather kidnapped because he's a direct descendant of the original Osea Father, Charington. Jess' cricket commentary then comes full circle as a new cycle emerges on the island: Sam attempts to escape in the rain, a flood leads him back to Osea, and the "Old Man" (aka the Father) sacrifices his life so that Sam can take his rightful spot as the island's leader.
Sam's three-episode arc ends with him reuniting with his son at the "Big House." Meanwhile, crickets swarm outside the home and loom in masses over the island. Given that Sam's son Nathan has a history of torturing insects, the final images may imply that father and son are united by a dark energy that's specific to their Osea heritage. Together, they may have the power to harness and re-direct negative energy (the crickets) and thus keep the island pure. On a deeper level, there's an argument to be made that Sam died on Osea and spiritually returned to the "Big House" where he could be with his son. In that case, The Third Day on HBO may prove to be a more complex character study about a grieving father's descent into the darkest depths of his mind. However, it's worth noting that the final three episodes featuring Naomie Harris do indeed include many of the same supporting characters. If Sam shows up, then it's safe to assume that the "Big House" is indeed a real location.