Affinity maturation, the process in which somatic hypermutation and positive selection generate antibodies with increasing affinity for an antigen, is pivotal in acquired humoral immunity. We have studied the mechanism of affinity gain in a human B-cell lineage in which two main maturation pathways, diverging from a common ancestor, lead to three mature antibodies that neutralize a broad range of H1 influenza viruses. Previous work showed that increased affinity in the mature antibodies derives primarily from stabilization of the CDR H3 loop in the antigen-binding conformation. We have now used molecular dynamics simulations and existing crystal structures to identify potentially key maturation mutations, and we have characterized their effects on the CDR H3 loop and on antigen binding using further simulations and experimental affinity measurements, respectively. In the two maturation pathways, different contacts between light and heavy chains stabilize the CDR H3 loop. As few as two single-site mutations in each pathway can confer substantial loop stability, but none of them confers experimentally detectable stability on its own. Our results support models of the germinal center reaction in which two or more mutations can occur without concomitant selection and show how divergent pathways have yielded functionally equivalent antibodies.