We study the charmless decays B -> Lambda(Lambda) over barh , where h stands for pi(+), K+, K-0, K*(+), or K*(0), using a 605 fb(-1) data sample collected at the Gamma(4S) resonance with the Belle detector at the KEKB asymmetric energy e(+)e(-) collider. We observe B-0 -> Lambda(Lambda) over barK(0) and B-0 -> Lambda(Lambda) over barK*(0) with branching fractions of (4.76(-0.68)(+0.84) (stat) +/- 0. 61 (syst)) x 10(-6) and (2.46(-0.72)(+0.87) +/- 0.34) x 10(-6), respectively. The significances of these signals in the threshold-mass enhanced mass region, M Lambda(Lambda) over bar < 2.85 GeV / c(2), are 12.4 sigma and 9.3 sigma, respectively. We also update the branching fraction B(B+ -> Lambda<(Lambda)over bar>K+) = (3.38(-0.36)(+0.41) +/- 0.41) x 10(-6) with better accuracy, and report the following measurement or 90% confidence level upper limit in the threshold-mass-enhanced region: B(B+ -> Lambda(Lambda) over barK*(+)) = (2.19(-0.88)(+1.13) +/- 0.33) x 10(-6) with 3.7 sigma significance; B(B+ -> Lambda(Lambda) over bar pi(+)) < 0.94) x 10(-6). A related search for B-0 -> Lambda<(Lambda)over bar>(D) over bar0 yields a branching fraction B(B-0 -> Lambda(Lambda) over bar(D) over bar (0)) = (1.05(-0.44)(+0.57) +/- 0.14) x 10(-5). This may be compared with the large, similar to 10(-4), branching fraction observed for B-0 -> p (p) over bar(D) over bar (0). . The M Lambda(Lambda) over bar enhancements near threshold and related angular distributions for the observed modes are also reported.