Passing string to a function in C - with or without pointers?

The accepted convention of passing C-strings to functions is to use a pointer:

void function(char* name)

When the function modifies the string you should also pass in the length:

void function(char* name, size_t name_length)

Your first example:

char *functionname(char *string name[256])

passes an array of pointers to strings which is not what you need at all.

Your second example:

char functionname(char string[256])

passes an array of chars. The size of the array here doesn't matter and the parameter will decay to a pointer anyway, so this is equivalent to:

char functionname(char *string)

See also this question for more details on array arguments in C.