Designers may often ask themselves how to adjust their design concepts to achieve demanding functional goals. To answer such questions, designers must often consider counterfactuals, weighing design alternatives and their projected performance. This paper introduces Multi-objective Counterfactuals for Design (MCD), a computational tool that automates and streamlines the counterfactual search process and recommends targeted design modifications that meet designers' unique requirements. MCD improves upon existing counterfactual search methods by supporting multi-objective requirements, which are crucial in design problems, and by decoupling the counterfactual search and sampling processes, thus enhancing efficiency and facilitating objective trade-off visualization. The paper showcases MCD's capabilities in complex engineering tasks using three demonstrative bicycle design challenges. In the first, MCD effectively identifies design modifications that quantifiably enhance functional performance, strengthening the bike frame and saving weight. In the second, MCD modifies parametric bike models in a cross-modal fashion to resemble subjective text prompts or reference images. In a final multidisciplinary case study, MCD tackles all the quantitative and subjective design requirements introduced in the first two problems, while simultaneously customizing a bike design to an individual rider's biomechanical attributes. By exploring hypothetical design alterations and their impact on multiple design objectives, MCD recommends effective design modifications for practitioners seeking to make targeted enhancements to their designs. The code, test problems, and datasets used in the paper are available to the public at decode.mit.edu/projects/counterfactuals/.