design something for someone
to conceive of something for someone; to draw up plans of something for someone. Ann designed a new kitchen for us. Would you design a book cover for me?
design something for something
to conceive of something that is to be part of something else; to draw up plans for something that is to be part of something else. The engineers designed a cure for the rattling problem. Bob, the computer programmer, designed a fix for the bug in the program. She designed a system of shelving for the library.