Recently a repeating fast radio burst (FRB) 121102 has been confirmed to be an extragalactic event and a persistent radio counterpart has been identified. While other possibilities are not ruled out, the emission properties are broadly consistent with Murase et al. (2016) that theoretically proposed quasi-steady radio emission as a counterpart of both FRBs and pulsar-driven supernovae. Here we constrain the model parameters of such a young neutron star scenario for FRB 121102. If the associated supernova has a conventional ejecta mass of $M_{rm ej}gtrsim{rm a few} M_odot$, a neutron star with an age of $t_{rm age} sim 10-100 rm yrs$, an initial spin period of $P_{i} lesssim$ a few ms, and a dipole magnetic field of $B_{rm dip} lesssim {rm a few} times 10^{13} rm G$ can be compatible with the observations. However, in this case, the magnetically-powered scenario may be favored as an FRB energy source because of the efficiency problem in the rotation-powered scenario. On the other hand, if the associated supernova is an ultra-stripped one or the neutron star is born by the accretion-induced collapse with $M_{rm ej} sim 0.1 M_odot$, a younger neutron star with $t_{rm age} sim 1-10$ yrs can be the persistent radio source and might produce FRBs with the spin-down power. These possibilities can be distinguished by the decline rate of the quasi-steady radio counterpart.
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