This is a very hard question, because there is so much we do not know about texts in the ancient world. Here are a few considerations.
First, there are internal and external evidence that we need to consider. Based on the internal evidence, the person claims to be Jude, the brother of James. The difficulty here is that both James and Jude are popular names. So, we cannot be certain who these people are. If we assume that James was the brother of Jesus, then Jude would also be the brother of Jesus as well. This is a possibility, but we cannot be certain, especially because he does not make mention of this fact. Also from internal evidence, we can say that he is not an apostle, as he distances himself from them in 1:17.
Second, from external evidence, all the false teachings and teachers seem to suggest a later date. In other words, to fight heresy, you need to first establish orthodoxy. So, some might believe that these letter were written later by another Jude about whom we do not know too much. This, too, is possible.
In conclusion, we probably should be agnostic about authorship, but if we had to take a guess, it is probably Jude, the brother of Jesus.
The name Jude is another form of the Hebrew name Judah or Judas. (See Thayer's Greek Lexicon for more information; link included.) This name appears several times in the New Testament, and there are several theories about the authorship of this book. The most commonly suggested authors are Judas the Apostle, who was also known as Thaddeus (see Matthew 10:2-4, Mark 3:18) and Judas, the half-brother of Jesus.
There is much evidence supporting the idea that the latter Judas was the author of Jude. Firstly, Jude introduces himself as "brother of James." Jude's brother James was not only the author of the book of James, but he was also the leader of the church in Jerusalem. Jude probably included this relationship in the first verse because his brother was so well-known among believers.
Another reason people believe his authorship is found in Jude 17. In this verse, Jude says, "But you, beloved, ought to remember the words that were spoken beforehand by the apostles of our Lord Jesus Christ..." Jude does not seem to refer to himself as an apostle, leading us to believe that it was not Judas the Apostle who wrote the book.