  We report an experimental study of the longitudinal relaxation time ($T_1$)
of the electron spin associated with single nitrogen-vacancy (NV) defects
hosted in nanodiamonds (ND). We first show that $T_1$ decreases over three
orders of magnitude when the ND size is reduced from 100 to 10 nm owing to the
interaction of the NV electron spin with a bath of paramagnetic centers lying
on the ND surface. We next tune the magnetic environment by decorating the ND
surface with Gd$^{3+}$ ions and observe an efficient $T_{1}$-quenching, which
demonstrates magnetic noise sensing with a single electron spin. We estimate a
sensitivity down to $\approx 14$ electron spins detected within 10 s, using a
single NV defect hosted in a 10-nm-size ND. These results pave the way towards
$T_1$-based nanoscale imaging of the spin density in biological samples.
