Little Father Time doesn't show up until really late in the novel, but boy does he make his role count. From the moment we meet the child of Jude and Arabella, we know something is a little off. He gets his nickname because he is "Age masquerading as Juvenility" (5.3.42)—in other words, Little Father Time may be biologically a child, but spiritually, the kid is as old as Jiroemon Kimura.
Little Father Time carries an unshakeable sadness with him that seems totally out of keeping with his young age. In spite of his love for Sue and his affection for Jude (whom he calls Mother and Father respectively), he just can't seem to find happiness in anything around him:
'I should like the flowers very much, if I didn't keep thinking they'd all be withered in a few days!' (5.5.87).
In spite of the fact that Little Father Time keeps talking about how miserable everything makes him, we did not actually expect him to turn to murder in Part Sixth, Chapter 2. And in fact, Hardy does not show us the actual scene of his murder of his younger siblings, nor does he give us direct insight into LFT's thought processes—though we do get a warning sign, when LFT tells Sue before she leaves to meet Jude that, "If we children were gone there'd be no trouble at all" (6.2.31).
LFT is a smart kid, and he can see that Sue and Jude are struggling financially and socially. Sue explains to him (from her own rather depressed perspective) that sometimes people don't like to rent rooms to families with children, which is why she and Jude are struggling to find a place for all of them to live.
But somehow, in spite of all of these warning signs, we still find the scene of Jude and Sue's discovery of the three dead children absolutely shocking. What adds to our sense of horror is LFT's pitiful suicide note: "Done because we are too menny" (6.2.40). His misspelling of "many" only reminds us how young LFT is when he makes this terrible, irreversible decision.
But what really gets us about this whole scene is that LFT clearly, genuinely believes that he is helping Jude and Sue. His loves his parents, and he wants to improve their lives. He has killed himself and his siblings because they are "too menny"—they are a burden on their parents and, logically (according to LFT's horrifying way of thinking), if there are fewer children, then Jude and Sue will be able to take care of themselves better.
The cruel irony that LFT does the worst thing in the novel thinking that it is the best course of action is one of the things that has given Jude the Obscure a justifiable reputation for extreme grimness. How could anything be worse than this? That Hardy is one grim fella.
Little Father Time works as a (deeply grim) plot device to kill Jude and Sue's two babies and himself, therefore starting the downward spiral that ultimately leads to Jude and Sue's deaths. However, he is also in the novel to foreshadow what Hardy sees coming in the future if life in general doesn't start improving:
'It was in [Little Father Time's] nature to do it. The doctor says there are such boys springing up amongst us—boys of a sort unknown in the last generation—the outcome of new views of life. They seem to see all its terrors before they are old enough to have staying power to resist them. He says it is the beginning of the coming universal wish not to live' (6.2.43)
In Little Father Time, we see the hints of a boy who is, for all intents and purposes, clinically depressed. He truly believes he never should have been born. But while today we might say that this boy needs some kind of intervention from a doctor, Hardy is writing at a time long before clinical services for mental health had become as varied and sophisticated as they are today. So when the doctor mentioned in this passage hears about Little Father Time, he diagnoses the boy as the symptom of a social rather than a medical problem.
For the doctor, there is a whole generation of boys facing "new views of life"—new, modern ways of living—who have had to confront life in "all its terrors" before they have the emotional or mental resilience to deal with it. Little Father Time is Hardy's grim view of things to come: as life becomes more and more unrelentingly cruel to good people, no one's going to want to go on and there is going to be a "universal wish not to live." You can't get a vision of the future much more dire than this.