Biological systems with their complex biochemical networks are known to be intrinsically noisy. Here we investigate the dynamics of actin polymerization of amoeboid cells, which are close to the onset of oscillations. We show that the large phenotypic variability in the polymerization dynamics can be accurately captured by a generic nonlinear oscillator model in the presence of noise. We determine the relative role of the noise with a single dimensionless, experimentally accessible parameter, thus providing a quantitative description of the variability in a population of cells. Our approach, which rests on a generic description of a system close to a Hopf bifurcation and includes the effect of noise, can characterize the dynamics of a large class of noisy systems close to an oscillatory instability.