Harper Lee had a good ear—this is typical educated speech in the US south, mixing formal syntax with very relaxed colloquial diction.
For line see ODO, s.v. line, 4.2:
A connected series of people following one another in time (used especially of several generations of a family):
we follow the history of a family through the male line.
ODO is less helpful with Lee's use of run here, but these three uses bracket the sense:
3.5 [NO OBJECT] (run in) (Of a quality, trait, or condition) be common or inherent in members of (a particular family), especially over several generations:
weight problems run in my family
3.6 [NO OBJECT] Pass into or reach a specified state or level:
inflation is running at 11 percent
[WITH COMPLEMENT]: the decision ran counter to previous government commitments
6.1 [NO OBJECT] (Of a system, organization, or plan) operate or proceed in a particular way:
everything’s running according to plan
With an adjectival predicate complement we might generalize these as something like "exhibit the specified quality throughout its course".
High to daughters marries two sorts of complement to run. High is an ordinary adjectival predicate complement, as when we say that a river is running "high"—that is, the water level is higher than normal. With run the to in to daughters doesn't designate a goal but a path; we say for instance My taste runs more to Joyce than to James, meaning that I tend to prefer James Joyce's writing to Henry James'—the spatial metaphor here is that the "course" of my reading runs more often through Joyce's writing.
The passage in question, then, could be paraphrased
He established a family which in each generation exhibited an unusually high proportion of daughters.