A few observations on some patterns in the play:

1. Both Hamlet and Ophelia (male and female, respectively) are imposed upon by their only surviving parent (female and male, respectively) in the first scene that either shares with her or with him. Hamlet is prevented from returning to Wittenburg first by Claudius but this request is reiterated then again by Gertrude, to whom the prince makes his reply. Ophelia is twice made to promise that she will not see Hamlet again, first to her brother, Laertes, and then again to her father. Fittingly, both Hamlet and Ophelia - within the span of two back-to-back scenes - make nearly the same (possibly ironic) reply:

Let not thy mother lose her prayers, Hamlet.
I pray thee, stay with us. Go not to Wittenburg.

I shall in all my best obey you, madam

compare to:

{long-winded speech trimmed}
I would not, in plain terms, from this time forth
Have you slander any moment leisure
As to give words or talk with the Lord Hamlet.
Look to 't, I charge you. Come your ways.

I shall obey, my lord.

2. Hamlet sure likes to say "adieu" quite a bit. Perhaps it's because his father's closing words to him before vanishing into the air are:

Fare thee well at once!
The glow-worm shows the matin to be near,
And 'gins to pale his uneffectual fire:
Adieu, adieu! Hamlet, remember me.

which Hamlet then repeats in paraphrase:

So, uncle, there you are. Now to my word;
It is 'Adieu, adieu! remember me.'
I have sworn 't.

Or perhaps the sound of the word is just in the prince's subconscious. He does ask, in his first soliloquy - before he meets his father's ghost, for his "too, too-solid flesh" to "melt, / Thaw and resolve itself into a dew," (I.ii.133-4) after all.

In any case, he certainly loves to repeat the sound. He uses it to close the letter that he addresses to Ophelia and that Polonius reads aloud to the King and Queen:

'Doubt thou the stars are fire;
Doubt that the sun doth move;
Doubt truth to be a liar;
But never doubt I love.
'O dear Ophelia, I am ill at these numbers;
I have not art to reckon my groans: but that
I love thee best, O most best, believe it. Adieu.
'Thine evermore most dear lady, whilst
this machine is to him, HAMLET.'

Finally, and most infamously, he uses it in his final, bitter dismissal of Gertrude:

Heaven make thee free of it. I follow thee.--
I am dead, Horatio. --Wretched queen, adieu.--

And if that cold-blooded farewell isn't enough to discourage Freudian suggestion of maternal incest, then I don't know what is.

3. Everybody in the royal family (plus a few) dies of poison:
  • Hamlet (poisoned with a lance)
  • Hamlet Sr. (poisoned in the ear by Claudius)
  • Gertrude (drinks from a poisoned cup)
  • Claudius (killed by the lance that poisons Hamlet, then forced to drink from Gertrude's cup)
  • Laertes (killed by his own poisoned sword)
Fitting for a play that deals so intensely with disease and corruption, no?

4. Check this out! And this and this!