Little Father Time is Jude and Arabella's son. He is called Little Father Time because, although he is young in body, his spirit is old. He has a strong sense of time as transitory, meaning he is sad that nothing stays the same. For instance, he mourns that he can't enjoy the flowers in their full bloom because all he can see is that they will soon die.
Little Father Time simply can't live in the present: he is always looking to the future, and he is always seeing it as depressing. He represents a grim, almost hopeless outlook on life. Nothing will get better, he thinks. It will, however, get worse.
This mindset causes him to kill himself and Sue and Jude's children because he thinks there are too many of them and that they will be a burden. (To an extent, the pregnant Sue is also responsible for this for complaining bitterly about their economic woes to Father Time.)
The doctor treats Little Father Time as a representative of a new generation that doesn't want to live, saying he is one of many:
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.