This is by no means elegant, but if you want concise:
Ruby, 71 chars
def f(a)b=a[0];b[0,(0..b.size).find{|n|a.any?{|i|i[0,n]!=b[0,n]}}-1]end
If you want that unrolled it looks like this:
def f(words)
first_word = words[0];
first_word[0, (0..(first_word.size)).find { |num_chars|
words.any? { |word| word[0, num_chars] != first_word[0, num_chars] }
} - 1]
end