How to identify the orbital of the valence nucleon(s) of exotic nuclei is an important problem. The elastic magnetic electron scattering is an excellent probe to determine the valence structure of odd-A nuclei. The relativistic mean-field theory has been successfully applied to systematic studies of the elastic charge electron scattering from even-even exotic nuclei. The extension of this method to investigate the elastic magnetic electron scattering from odd-A exotic nuclei is a natural generalization. The experimental form factors of O17 and Ca41 are reproduced very well with the help of the spectroscopic factors which are introduced into the relativistic treatment of the magnetic electron scattering for the first time. The emphases are put on the magnetic form factors of C15,17,19, O23, F17, and Ca49,59 calculated in the relativistic impulse approximation. Great differences have been found in the form factors of the same nucleus with different configurations. Therefore, the elastic magnetic electron scattering can be used to determine the orbital of the last nucleon of odd-A exotic nuclei. Our results can provide references for the electron scattering from exotic nuclei in the near future.