How do I calculate a good hash code for a list of strings?

Your first option has the only inconvenience of (String1, String2) producing the same hashcode of (String2, String1). If that's not a problem (eg. because you have a fix order) it's fine.

"Cat all the string together then get the hashcode" seems the more natural and secure to me.

Update: As a comment points out, this has the drawback that the list ("x", "yz") and ("xy","z") would give the same hash. To avoid this, you could join the strings with a string delimiter that cannot appear inside the strings.

If the strings are big, you might prefer to hash each one, cat the hashcodes and rehash the result. More CPU, less memory.

I see no reason not to concatenate the strings and compute the hashcode for the concatenation.

As an analogy, say that I wanted to compute a MD5 checksum for a memory block, I wouldn't split the block up into smaller pieces and compute individual MD5 checksums for them and then combine them with some ad hoc method.

Standard java practise, is to simply write

final int prime = 31;
int result = 1;
for( String s : strings )
{
    result = result * prime + s.hashCode();
}
// result is the hashcode.