Question:

Why in the Bible does Jesus refer to himself as the Son of man?

My answer:

Jesus says (in the Gospel according to John chapter 5):

21   For just as the Father raises the dead and gives them life, even so the
     Son gives life to whom he is pleased to give it.
22   Moreover, the Father judges no one, but has entrusted all judgment to
     the Son,
23   that all may honor the Son just as they honor the Father. He who does
     not honor the Son does not honor the Father, who sent him.

...

26   For as the Father has life in himself, so he has granted the Son to
     have life in himself.
27   And he has given him authority to judge because he is the Son of Man.
                                             ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The final judgement [i.e. the judgment of Judgment Day] is in the hand
of Jesus according to his own words. Is that claim enough to his status
as Deity?

"Son of Man" seems to be a pretty important title. A more detailed look 
at the meaning of this title you can find under the articles on the 
Trinity, part 1 on the deity of Christ. 
A vision and prophecy of Prophet Daniel is the source of this title. 
And observe how many attributes are given in the same way to God and to 
the one like a son of man. There seems to be some intimate relationship.



Please keep in mind that I am a fallible human being and I am trying
to answer to the best of my knowledge -- which is not perfect. I do not 
claim to be right in all I say. I am always glad about corrections 
to my answers or a renewed question that asks for more clarification
where I stayed unclear. 



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