StringCases[#, ___ ~~ "Distribution" ~~ ___ :> #] & /@ Names["System`*"]; DeleteCases[%, {}]
Gives
{{"ArcSinDistribution"}, {"BarabasiAlbertGraphDistribution"}, {"BatesDistribution"}, {"BeckmannDistribution"}, \ {"BenfordDistribution"}, {"BeniniDistribution"}, {"BenktanderGibratDistribution"}, {"BenktanderWeibullDistribution"}, \ {"BernoulliDistribution"}, {"BernoulliGraphDistribution"}, {"BetaBinomialDistribution"}, {"BetaDistribution"}, \ {"BetaNegativeBinomialDistribution"}, {"BetaPrimeDistribution"}, {"BinomialDistribution"}, {"BinormalDistribution"}, \ {"BirnbaumSaundersDistribution"}, {"BorelTannerDistribution"}, {"CauchyDistribution"}, {"CensoredDistribution"}, \ {"ChiDistribution"}, {"ChiSquareDistribution"}, {"CompoundPoissonDistribution"}, {"CopulaDistribution"}, {"CoxianDistribution"}, \ {"DagumDistribution"}, {"DataDistribution"}, {"DavisDistribution"}, {"DegreeGraphDistribution"}, {"DirichletDistribution"}, \ {"DiscreteUniformDistribution"}, {"DistributionChart"}, {"DistributionDomain"}, {"DistributionFitTest"}, \ {"DistributionParameterAssumptions"}, {"DistributionParameterQ"}, {"EmpiricalDistribution"}, {"ErlangDistribution"}, \ {"EstimatedDistribution"}, {"ExpGammaDistribution"}, {"ExponentialDistribution"}, {"ExponentialPowerDistribution"}, \ {"ExtremeValueDistribution"}, {"FailureDistribution"}, {"FindDistributionParameters"}, {"FirstPassageTimeDistribution"}, \ {"FisherHypergeometricDistribution"}, {"FisherZDistribution"}, {"FRatioDistribution"}, {"FrechetDistribution"}, \ {"GammaDistribution"}, {"GeometricDistribution"}, {"GompertzMakehamDistribution"}, {"GraphPropertyDistribution"}, \ {"GumbelDistribution"}, {"HalfNormalDistribution"}, {"HistogramDistribution"}, {"HotellingTSquareDistribution"}, \ {"HoytDistribution"}, {"HyperbolicDistribution"}, {"HyperexponentialDistribution"}, {"HypergeometricDistribution"}, \ {"HypoexponentialDistribution"}, {"InverseChiSquareDistribution"}, {"InverseGammaDistribution"}, {"InverseGaussianDistribution"}, \ {"JohnsonDistribution"}, {"KDistribution"}, {"KernelMixtureDistribution"}, {"KumaraswamyDistribution"}, {"LandauDistribution"}, \ {"LaplaceDistribution"}, {"LevyDistribution"}, {"LindleyDistribution"}, {"LogGammaDistribution"}, {"LogisticDistribution"}, \ {"LogLogisticDistribution"}, {"LogMultinormalDistribution"}, {"LogNormalDistribution"}, {"LogSeriesDistribution"}, \ {"MarginalDistribution"}, {"MaxStableDistribution"}, {"MaxwellDistribution"}, {"MeixnerDistribution"}, {"MinStableDistribution"}, \ {"MixtureDistribution"}, {"MoyalDistribution"}, {"MultinomialDistribution"}, {"MultinormalDistribution"}, \ {"MultivariateHypergeometricDistribution"}, {"MultivariatePoissonDistribution"}, {"MultivariateTDistribution"}, \ {"NakagamiDistribution"}, {"NegativeBinomialDistribution"}, {"NegativeMultinomialDistribution"}, {"NoncentralBetaDistribution"}, \ {"NoncentralChiSquareDistribution"}, {"NoncentralFRatioDistribution"}, {"NoncentralStudentTDistribution"}, {"NormalDistribution"}, \ {"OrderDistribution"}, {"ParameterMixtureDistribution"}, {"ParetoDistribution"}, {"PascalDistribution"}, {"PearsonDistribution"}, \ {"PERTDistribution"}, {"PoissonConsulDistribution"}, {"PoissonDistribution"}, {"PolyaAeppliDistribution"}, {"PowerDistribution"}, \ {"PriceGraphDistribution"}, {"ProbabilityDistribution"}, {"ProductDistribution"}, {"RayleighDistribution"}, \ {"ReliabilityDistribution"}, {"RiceDistribution"}, {"SechDistribution"}, {"SinghMaddalaDistribution"}, {"SkellamDistribution"}, \ {"SkewNormalDistribution"}, {"SliceDistribution"}, {"SmoothKernelDistribution"}, {"SpatialGraphDistribution"}, \ {"SplicedDistribution"}, {"StableDistribution"}, {"StandbyDistribution"}, {"StationaryDistribution"}, {"StudentTDistribution"}, \ {"SurvivalDistribution"}, {"SuzukiDistribution"}, {"TransformedDistribution"}, {"TriangularDistribution"}, \ {"TruncatedDistribution"}, {"TsallisQExponentialDistribution"}, {"TsallisQGaussianDistribution"}, {"TukeyLambdaDistribution"}, \ {"UniformDistribution"}, {"UniformGraphDistribution"}, {"UniformSumDistribution"}, {"VarianceGammaDistribution"}, \ {"VoigtDistribution"}, {"VonMisesDistribution"}, {"WakebyDistribution"}, {"WalleniusHypergeometricDistribution"}, \ {"WaringYuleDistribution"}, {"WattsStrogatzGraphDistribution"}, {"WeibullDistribution"}, {"WignerSemicircleDistribution"}, \ {"ZipfDistribution"}}