A lettuce cultivar, designated 51-0501050-B, is disclosed. The invention relates to the seeds of lettuce cultivar 51-0501050-B, to the plants of lettuce cultivar 51-0501050-B and to methods for producing a lettuce plant by crossing the cultivar 51-0501050-B with itself or another lettuce cultivar. The invention further relates to methods for producing a lettuce plant containing in its genetic material one or more transgenes and to the transgenic lettuce plants and plant parts produced by those methods. This invention also relates to lettuce cultivars or breeding cultivars and plant parts derived from lettuce cultivar 51-0501050-B, to methods for producing other lettuce cultivars, lines or plant parts derived from lettuce cultivar 51-0501050-B and to the lettuce plants, varieties, and their parts derived from the use of those methods. The invention further relates to hybrid lettuce seeds, plants, and plant parts produced by crossing cultivar 51-0501050-B with another lettuce cultivar.