‘And Arpachshad begat Shelah, and Shelah begat Eber and to Eber were born two sons. The name of the one was Peleg, for in his days the earth was divided, and his brother's name was Joktan.'
The movement from ‘begat' to ‘born two sons' is in reverse to the earlier ‘sons of' becoming ‘begat' (Genesis 10:8). The inclusion of both descriptions in the same sentence, yet in different order, serves to demonstrate that such changes are purely stylistic and not evidence of separate narratives.
“In his days the earth was divided”. Division is mentioned in Genesis 10:5 and Genesis 10:32, but there the idea is of a gradual division into nations. ‘Division' is not mentioned in Genesis 11. The meaning may therefore be that ‘the land was divided' by irrigation channels. Peleg's name (‘water-courses, division') may have been given because of this very intention. We can compare Isaiah 30:25; Isaiah 32:2; Job 29:6; Job 38:25 where ‘peleg' means irrigation canals (Assyrian plagu). Alternately the ‘division' could refer to a dispute between the two sons, resulting in a divided land, like that between Abraham and Lot.
However it is possible that this is suggesting that Genesis 11 and the process of division into nations began at this time, but then why not more directly say that ‘that was when the people were scattered'?
The genealogy of Arpachshad at this point is clearly a genuine genealogy as we understand it (compare Genesis 11:10-14) as befits the ancestor of Eber and Abraham. Thus Peleg comes very early on in the period that produced the nations.
Eber - the name means ‘one who emigrates'. He is thought of as the eponymous ancestor of ‘the Hebrews', and the name appears to be referred to Israel in Numbers 24:24. But while the term ‘Hebrew' is referred to Abraham and his descendants, and very much later became in general use connected with the Jews, it was originally essentially used in a context where the term is applied by foreigners who saw them as immigrants and probably mainly used in derision. Israel did not see themselves as ‘the Hebrews'.