What criteria make an airport suitable for a technical stop for a large airliner?
Some factors might be: 


Price of fuel at the various airports?
Landing fees?
Are any of the airports so busy that you might have to hold waiting for landing clearance? (You don't want to plan a fuel stop at LHR).
Can you expect long taxis in some places? (All other things being equal, a single runway with a ramp next to it is better news than something like Schiphol or O'Hare).
If you're carrying passengers, are some of the airports in countries with significantly more cumbersome immigration rules applying to technical stops than others? Some may want airport transit visas from passengers of certain nationalities even if they don't disembark.
Political considerations such as the ones mentioned by Terry in the comment.
And, of course, the total expected fuel burn for the two legs, which will depend on your refueling points. Your operations department will make concrete calculations for a number of likely stops, and weather has an influence on top of geography. If you have several possible airports that all happen to be right on the shortest path between your origin and destination, the math will tend to favor a refueling stop roughly in the middle -- this will minimize the total cost-of-lifting-enough-fuel-for-the-rest-of-the-leg.


Finally, if you're planning a recurring scheduled flight, will any of the options provide a business opportunity to take on some extra passengers on days where the demand from the actual origin is low?