The thermal relic density of W-ino-like neutralino dark matter in brane world cosmology is studied. The expansion law at a high energy regime in brane world cosmology is modified from the one in standard cosmology, and the resultant relic density can be enhanced if the five-dimensional Planck mass M-5 is low enough. We calculate the W-ino-like neutralino relic density in the anomaly mediated supersymmetry breaking scenario and show that the allowed region is dramatically modified from the one in standard cosmology and the W-ino-like neutralino with mass of order 100 GeV can be a good candidate for dark matter. Since the allowed region disappears eventually as M-5 decreases, we can find a lower bound on M-5 greater than or similar to 100 TeV according to the neutralino dark matter hypothesis, namely, the lower bound in order for the allowed region of neutralino dark matter to exist.