“Evolution can’t be true because natural selection can’t account for everything.”
Apart from the first four words, the rest is absolutely correct. Natural selection is not the only evolutionary force. Genetic drift, mutation, and gene flow are also important mechanisms that do not depend on selection. (There are others that do depend on selection, but will not be discussed at this time)
Genetic drift works much the way it sounds: with allele frequencies randomly changing (drifting) from one generation to the next. There are several genotypic frequencies that can arise from a single set of allelic frequencies. For instance, even with equal representation, p(A)=q(a)=0.5, we could have a population with several homozygotes of AA and aa with few heterozygotes, or to begin with all heterozygotes and no homozygotes. Even with complete neutrality, allele frequencies (and thus genotype frequencies) can change over time. It is possible for one homozygous genotype to completely take over the population, in which case it has become “fixed” in the population. The only way to add genetic variety for that trait back into a closed population is through a mutation.
Mutations have the reputation for either always being bad (like in cancers) or unrealistically amazing (like in X-Men). Mutations can be deleterious, neutral, or advantageous. For instance, if a mutation occurs in a non-coding region of DNA and does not alter regulation or expression, it holds no benefit or disadvantage. Similarly, a mutation in a coding region that does not change the amino acid produced will also be completely neutral. Some mutations are beneficial, such as being heterozygous for Sickle Cell Trait, which provides effective malarial resistance. Being homozygous, however, results in Sickle Cell Anemia, and is highly deleterious.
Gene flow is the migration of reproductive individuals from one population to another. If you visited an isolated population on an island and effectively mated with its inhabitants, you may have either introduced new alleles into the population, or altered the frequency if you shared existing alleles. The extent of your influence would depend on the size of the population and the number of offspring you produced.
However powerful any of these forces may be on their own, it is important to remember that natural populations employ a combination of evolutionary mechanisms for change to occur over time.