First of all, you need an array, not a pointer.
static const char * const days[] = {"mon", "tue", "wed", "thur",
"fri", "sat", "sun"};
Second of all, you can’t initialize that directly inside the class definition. Inside the class definition, leave only this:
static const char * const days[]; //declaration
Then, in the .cpp file, write the definition
const char * const Week::days[] = {"mon", "tue", "wed", "thur",
"fri", "sat", "sun"};
Update for C++11
Now you can initialize members directly in the class definition:
const char * const days[] = {"mon", "tue", "wed", "thur",
"fri", "sat", "sun"};