Vaccine-based elicitation of broadly neutralizing antibodies holds great promise for preventing HIV-1 transmission. However, the key biophysical markers of improved antibody recognition remain uncertain in the diverse landscape of potential antibody mutation pathways, and a more complete understanding of anti-HIV-1 fusion peptide (FP) antibody development will accelerate rational vaccine designs. Here we survey the mutational landscape of the vaccine-elicited anti-FP antibody, vFP16.02, to determine the genetic, structural, and functional features associated with antibody improvement or fitness. Using site-saturation mutagenesis and yeast display functional screening, we found that 1% of possible single mutations improved HIV-1 envelope trimer (Env) affinity, but generally comprised rare somatic hypermutations that may not arise frequently in vivo. We observed that many single mutations in the vFP16.02 Fab could enhance affinity >1,000-fold against soluble FP, although affinity improvements against the HIV-1 trimer were more measured and rare. The most potent variants enhanced affinity to both soluble FP and Env, had mutations concentrated in antibody framework regions, and achieved up to 37% neutralization breadth compared to 28% neutralization of the template antibody. Altered heavy- and light-chain interface angles and conformational dynamics, as well as reduced Fab thermal stability, were associated with improved HIV-1 neutralization breadth and potency. We also observed parallel sets of mutations that enhanced viral neutralization through similar structural mechanisms. These data provide a quantitative understanding of the mutational landscape for vaccine-elicited FP-directed broadly neutralizing antibody and demonstrate that numerous antigen-distal framework mutations can improve antibody function by enhancing affinity simultaneously toward HIV-1 Env and FP.