We present a possible Cepheid-like luminosity estimator for the long gamma-ray bursts based on the variability of their light curves. To construct the luminosity estimator, we use CGRO/BATSE data for 13 bursts, Wind/KONUS data for 5 bursts, Ulysses/GRB data for 1 burst, and NEAR/XGRS data for 1 burst. Spectroscopic redshifts, peak fluxes, and high resolution light curves are available for 11 of these bursts; partial information is available for the remaining 9 bursts. We find that the isotropic-equivalent luminosities L of these bursts positively correlate with a rigorously-constructed measure V of the variability of their light curves. We fit a model to these data that accommodates both intrinsic scatter (statistical variance) and extrinsic scatter (sample variance). If one excludes GRB 980425 from the fit on the grounds that its association with SN 1998bw at a redshift of z = 0.0085 is not secure, the luminosity estimator spans approx. 2.5 orders of magnitude in L, and the slope of the correlation between L and V is positive with a probability of 1 - 1.4 x 10^-4 (3.8 sigma). Although GRB 980425 is excluded from this fit, its L and V values are consistent with the fitted model, which suggests that GRB 980425 may well be associated with SN 1998bw, and that GRB 980425 and the cosmological bursts may share a common physical origin. If one includes GRB 980425 in the fit, the luminosity estimator spans approx. 6.3 orders of magnitude in L, and the slope of the correlation is positive with a probability of 1 - 9.3 x 10^-7 (4.9 sigma). Independently of whether or not GRB 980425 should be included in the fit, its light curve is unique in that it is much less variable than the other approx. 17 light curves in our sample for which the signal-to-noise is reasonably good.
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