  We consider strings moving in the R_t x S^3_\eta subspace of the
\eta-deformed AdS_5 x S^5 and obtain a class of solutions depending on several
parameters. They are characterized by the string energy and two angular
momenta. Finite-size dyonic giant magnon belongs to this class of solutions.
Further on, we restrict ourselves to the case of giant magnon with one nonzero
angular momentum, and obtain the leading finite-size correction to the
dispersion relation.
