Jesse was the first child of Tom (Joseph Cooper's oldest child) and Lois Cooper. In Jesse's only scene, Tom is holding him and showing him to his grandfather, Joseph Cooper, via video transmission. Tom thought they could name him "Coop" because of his grandfather, but Lois thought it would be awkward. Donald declined having his great-grandson named after him.
Sadly, Jesse died from a lung ailment caused by the deteriorating conditions on Earth. He was buried on the Cooper farm near Erin Cooper (his grandmother) and Donald Cooper (his great-grandfather).
Later, Murph accuses Tom of waiting for his second child to die just like Jesse by staying in his home rather than vacating. This gives the idea of Jesse having died from a lung condition as Tom's second child also suffers from such an ailment.