Grammar for a language: odd length, middle character not repeated
Assume the middle character is $a$. Then a grammar is
$$A \rightarrow bAb | bAc | cAb | cAc | a$$
Similarly, we can define nonterminals $B$ and $C$ for the strings in our language where $b$ and $c$, respectively, are the middle symbols. Given those, you can get a grammar for your language by taking the union:
$$S \rightarrow A | B | C$$