The "longer" form is the only correct one.
Think of it this way: it's noun's birthday party. But noun doesn't have to just be a simple noun (like my son), it can also be a noun phrase (like the king of the moon or my son's friend), and 's will still go at the end. You would never say "friend birthday party" if it belongs to friend; it's always "friend's birthday party."
Don't be fooled by the fact that 's appears twice in "son's friend's birthday party"; the first one indicates that the friend belongs to your son, and the second one indicates that the party belongs to your son's friend. You can make this as long as you like - my uncle's barber's priest's next-door neighbor's grandson's roommate's girlfriend's sister's birthday party if you want - but the 's always has to get appended.