Doctor of Divinity.
The context is Jude ambitiously planning out his future life as an academic clergyman in Christminster (Oxford). From the same passage:
And then he continued to dream, and thought he might become even a bishop by leading a pure, energetic, wise, Christian life. [...]
["]Yes, Christminster shall be my Alma Mater; and I'll be her beloved son, in whom she shall be well pleased."
If he succeeds in getting into Christminster to study theology, he might well aspire to become a D.D. or Doctor of Divinity.
See also the following passage from "A Tragedy of Two Ambitions", a short story available here which bears some resemblances to Jude the Obscure and has been described as a "short story prelude" to the novel:
‘You’ll be a bishop, Joshua, before you have done!’
‘Ah!’ said the other bitterly, shaking his head. ‘Perhaps I might have been — I might have been! But where is my D.D. or LL.D.; and how be a bishop without that kind of appendage? Archbishop Tillotson was the son of a Sowerby clothier, but he was sent to Clare College. To hail Oxford or Cambridge as alma mater is not for me — for us! My God! when I think of what we should have been — what fair promise has been blighted by that cursed, worthless —’
Here again we see an ambitious youth referring to the D.D. and LL.D in his hope to become a successful academic at Oxford or Cambridge (here referred to by name rather than as part of Hardy's fictionalised Wessex).