Question

What does the line signify that "Genetic material must be able to express itself in the form of mendelian characters"?

Answer 1

That quote says that the genetic material should be able to write as Mendelian traits.
ie
Mendelian Traits are those traitswhich follow Mendel's rules of only 2 possible versions of a gene (1 dominant, 1 recessive).

In the case of genes we should be able to select dominant and recessive characters.