We present the results from our search for H I 21-cm absorption in a sample of 16 strong Fe II systems [W_r(Mg II λ2796) ≥ 1.0 Å and W_r(Fe II λ2600) or W_{Fe II} ≥ 1 Å] at 0.5 < z < 1.5 using the Giant Metrewave Radio Telescope and the Green Bank Telescope. We report six new H I 21-cm absorption detections from our sample, which have increased the known number of detections in strong Mg II systems at this redshift range by ∼50 per cent. Combining our measurements with those in the literature, we find that the detection rate of H I 21-cm absorption increases with W_{Fe II}, being four times higher in systems with W_{Fe II} ≥ 1 Å compared to systems with W_{Fe II} < 1 Å. The N(H I) associated with the H I 21-cm absorbers would be ≥2 × 10²⁰ cm^-2, assuming a spin temperature of ∼500 K (based on H I 21-cm absorption measurements of damped Lyman α systems at this redshift range) and unit covering factor. We find that H I 21-cm absorption arises on an average in systems with stronger metal absorption. We also find that quasars with H I 21-cm absorption detected towards them have systematically higher E(B - V) values than those which do not. Further, by comparing the velocity widths of H I 21-cm absorption lines detected in absorption- and galaxy-selected samples, we find that they show an increasing trend (significant at 3.8σ) with redshift at z < 3.5, which could imply that the absorption originates from more massive galaxy haloes at high z. Increasing the number of H I 21-cm absorption detections at these redshifts is important to confirm various trends noted here with higher statistical significance.