During the processing of sour cherries into different foodstuffs, a large amount of kernels is produced as waste material, which creates a significant disposal problem for the food industry. Sour cherry kernels containing 25.3–35.5% of protein can be used as a functional protein source in food production. Therefore, we aimed to study the effects of hydrolysis degree on the sour cherry kernel protein hydrolysates. Proteins were extracted from the defatted flour by isoelectric precipitation. The resulting protein concentrate was hydrolyzed (5, 10, and 15% hydrolysis) using Alcalase to yield hydrolysates. We determined their oil and water holding, emulsifying, gelation, and foaming properties, as well as apparent molecular weight distribution and proximate compositions. No protein fractions greater than an apparent molecular weight of about 22 kDa were present in the hydrolysates. The hydrolysis of the protein concentrate mostly led to an increase in protein solubility. As the degree of hydrolysis increased from 5 to 15%, the water holding capacity of the hydrolysates decreased from 2.50 ± 0.03 to 2.03 ± 0.02 g water/g, indicating its deterioration. The hydrolysates obtained at different degrees of hydrolysis had a better solubility than the intact protein concentrate. The oil holding capacity, the foaming stability, and the least gelation concentration of the protein concentrate could not be considerably improved by hydrolysis. In contrast, its emulsifying activity index and foaming capacity could be increased with a limited degree of hydrolysis (up to 10%).