I am one of those who avoids, in matters Shakespearean, the unique or gimmicky presentation. I find all such efforts distracting because Shakespeare's plays are so very familiar that any tinkering becomes misleading. That is not to say that every actor that has played a leading role, such as that of Hamlet, has not tried to put his own personal stamp on the character and a soliloquy, especially this one.
Lawrence Olivier, in the movie version, did the speech on a high precipice overlooking an angry sea while holding a dagger; Ethan Hawke did it in a kind of stupor while walking the aisles of a video store; Kenneth Baragh did it very dramatically while facing a mirror; Richard Buton presented it as a well-rehearsed speech as if Hamlet knew he was being overheard; Mel Gibsom over-dramatized it while walking in echoing catacombs. All too precious for my tastes.
What I would suggest is honesty to the words and the man. Treat it as a serious dialectical argument by a person who loves to think things out and to hear himself talk. It is a riff, an improvisation by a man who intends to do nothing but contemplate life and death for the sake of contemplating.
Take your time; weigh each argument as if it were a fresh and original idea... one thought arising oranically from the other. "To be, or not to be to be, that IS the question." Like that, as if you really meant to think it through. Take your time. Make a statement and answer it... let the thoughts unfold naturally.
Given all the "unique" ways this monumentally famous soliloquy has been presented over the centuries, present it quietly and honestly, as if it these were your thoughts and you were thinking them for the very first time. That may be the most unique presentation of all.
Listen to, but don't look at (for it's just too creepy), the YouTube link below, for it is something very close to what I am suggesting.
Jude Law is doing Hamlet on Broadway, and he delivered the soliloquy in a literal blizzard to symbolize his coldness and doubt. So, you will need several ice machines and warm clothes for that one.
My favorite renditions are the parodies:
The Student's Soliloquy
To slack, or not to slack, that is the question;
Whether 'tis nobler in the mind to suffer
The zeros and low marks of outrageous assignments,
Or to take pens against a sea of compositions,
And by opposing, finish them.
The Stock Broker's Soliloquy
To buy or not to buy, that is the decision;
Whether ‘tis nobler in the mind to suffer
The ups and downs of the unpredictable stock market,
Or to take arms against a sea of merciless profiteers
And by getting the inside news, become a millionaire.
You might start with an unusual sort of costume. Perhaps something modern? Since he's really considering his own life it would be interesting to do portions of the monologue standing in front of a mirror, truly "self-reflecting." You might even alternate between different mirrors, using both a handheld and a full-length mirror. If you set up an empty frame you could even face the audience and be looking straight out at them as you appear to gaze into a mirror. If you choose to modernize it, you could even carry a prescription bottle symbolizing his struggle with living or dying.