We can use patchelf:
patchelf --replace-needed liboriginal.so.1 libreplacement.so.1 my-program
We can also remove a dependency:
patchelf --remove-needed libfoo.so.1 my-program
Add a dependency:
patchelf --add-needed libfoo.so.1 my-program
Or change the path where to search for the libraries (rpath):
patchelf --set-rpath /path/to/lib:/other/path my-program