Things to Consider

Some of these may have no relevance to you and some answers may be hard to find but I'd recommend having a read through and see if any of them make it into your must have box. You never know what will be the detail that makes the trip.

Points to consider Comments
Are you given a detailed itinerary? Do not trust and seriously question any itinerary that guarantees any particular landing. All Arctic trips are entirely at the mercy of weather and ice conditions. While you might get a list of planned or hoped for spots, everything is subject to change at a moment's notice and the captain has complete discretion to cancel or cut any activity short due to weather concerns. 
What's the average passenger like? Highly subjective and highly variable - but what are the normal ship demographics? Is there an average age and nationality of passengers? Generally speaking the more expensive and luxury ships trend older, more spartan, younger and/or focussed on a particular aim ie lots of photographers. Suites and ships with double beds will be less suited to the solo traveller. If the operating company is based or has booking offices in a certain location then expect more people of this nationality. 
Your personal fitness Hard one to answer but worth thinking about. The Arctic is, generally speaking, pretty low energy. You need some level of fitness to get on and off zodiacs, but once you're on shore there will always be a variety of options for levels of activity. However, if you cannot cope with stairs, this may not be the destination for you. If you have a medical condition that has the potential to need access to a fully equipped medic, think twice before booking. Every ship will have a doctor, but they're usually volunteering time and are planning for seasickness, sprains and possibly broken bones. Getting to either a clinic or a fully equipped hospital will take several days and involve diverting the entire ship and cancelling all landings and activities for the rest of the passengers. There's a small hospital in Longyearbyen, but full medical facilities will require evacuation to the mainland of Norway.
What are you expecting? You're on this site which is good, you're doing research! Read trip reports - are you expecting a cruise where you get up at your leisure, eat in one of the many restaurants and play in the casino before going to see the polar bear that will be sitting nicely on an ice floe under your balcony? If you're planning for 7am wake up calls every morning, breakfast, landing, lecture, lunch, lecture, zodiac cruise, snack, recap, dinner, then you're probably more on course to have expectations met. Get whoever you book with to explain a typical day so you understand what your ship will provide.
Expedition staff Very subjective, but does everyone have a relevant qualification, how many years experience, good lecturing skills? Staff to passenger ratio? How much do they interact and how available are they for guests. Mix of nationalities or mostly from one country? Other languages spoken?
Hotel staff Very subjective, but well regarded efficient service? Staff to passenger ratio. Level of cabin service? Can range from 24hr butler, to a nice girl who leaves you origami towel animals once a day.
Ship crew Very subjective but are the crew experienced? Do they have a qualified ice pilot? What's the ship and company's safety record like?
AECO member Arctic Expedition Cruise Operators - Have they signed up to agree to preserve the Arctic and if not, why not?
Groups Are there any large groups of a single nationality/interest on your departure - ie 35 Estonian bird watching photographers may change the dynamic of the ship and how landings operate. Groups almost always have cabins and berths reserved well in advance so you should be able to find out on booking if this is important to you.
Families Are children welcomed on board or do you want an adult only ship? If you're travelling with your family are there special activities/meals for your children? Is any gear offered in smaller sizes?
Practicalities How do you settle your bill? what currencies are accepted? What are the plugs? Is there a safe in the room and door locks? How do cabin charges work? What's expected in terms of gratuities? Is there a medic and what do they charge?
After travel Do you get a post voyage DVD/USB stick with a slideshow, trip notes, copies of photographs? Are they included or is there an extra cost?
Pre and post departure Are hotels included and airport transfers? When is the latest you should arrive and earliest flight you should book home? What are luggage weight limits and how does the ship cope with lost or delayed bags? Do they or you provide evacuation insurance?

Does the ship have...?
A mud room For storage and drying of your salt coated clothing and saves space in your cabin.
Side zodiac access Good so the ship can create a lee in case of high seas
Stern zodiac access Potentially easier and more stable platform
Theme evenings Dress like a walrus! Make a hat out of towels! Hide in your cabin and pretend no-one's in costume!
Al fresco dining For the occasional bbq on deck or more routine lunches
Entertainment on board Musician? Crew band? Grand piano in the lounge? TV/DVD in the cabin? Movie nights in the lounge? Board games, library? Eye spy something beginning with I.
Connectivity Wifi/Internet packages, Satphone, email, carrier pigeon. Note: All internet is satellite based and so will be expensive, slow and with patchy coverage so even if you buy a package if weather is inclement or you're in the lee of a fjord there may be no signal. Fine for sending the occasional text based message to friends and family, and potentially updating facebook, but no good for almost anything else.
Alcohol Included as part of a package or what's the cost if not
Gym Exercise with a view - what equipment is there?
Deck access Can you walk all the way around the ship? Is there a high 360 observation deck? Can you access the bow and stern? Is there an open bridge policy? Is there room for all passengers to be standing at a railing with views when a bear is sighted?
Jacuzzi/pool/sauna In case you like your water warmer rather than ice studded
Single cabins For those who don't want to share or pay a supplement
Quad cabins Lots of people in a small space for a lower price. As you're mostly only in the cabin to sleep, this can be an excellent value option and a good way to make new friends.
Meal options Buffet or menu based meals? How many selections at each meal? Are you happy to mingle on large tables with fellow passengers? Do the expedition staff eat and chat with you over meals? Do you want a table to yourself? Can the ship cater for your dietary requirements? How many dining places are there? Can everyone eat at once? Does the dining room have a view? Is the food midrange cafeteria or high end restaurant quality?
Included items so you don't have to pack Do you get given or loaned a Parka and boots? What are the toiletries in cabins and is there a hairdryer? Does the ship have a supply of walking sticks to borrow?
Helicopters For access to hard to reach locations - usually only on trips that are trying to get to the Pole as use of helicopters for tourism is banned in Svalbard.
Accessibility adaptations How do you get around the ship? Are there lifts? What's the number of stairs to get between your cabin and common areas? How do you access the zodiacs? Can the ship cater for specialised needs be they mobility or other medical?
Public areas Can everyone get to presentations at once? How many meal sittings are there? Are there indoor viewing lounges? Is the seating comfy looking? Can you plug your laptop in in the lounge?
Gift shop What's in it - essentials or just souvenirs and when is it open?
Zodiac/Polar Cirkels How do you get to shore? Can everyone on board be off the ship at once for zodiac cruising?