Any given small dog would suit your needs; each dog is an individual and so it depends on the training and temperament of the specific dog. Given that you don't need anything *too* particular, your family would do well by going to a local shelter or rescue. You all can meet with the staff there, tell them what you're looking for in a dog, and they can match you up with the best candidates to fit your lifestyle.
If you get a dog JUST for the lap-dog factor, you'll easily be disappointed. You should get a dog to be a DOG and if you go into it expecting something particular (such as an extremely cuddly dog), you're setting yourselves up for disappointment. This is how many dogs get surrendered to shelters--because a family wanted a dog that was snuggly, non-energetic, already trained, etc. and that is NEVER guaranteed. Get a dog to have as a fun companion, and re-think this whole dog thing if you can't get the "lap-dog" thing off of your minds.
Since you mentioned trainability, you can't get any more intelligent than a poodle. You could get a mini or toy, as long as you're willing to deal with the extensive grooming requirements. Your kids will have a lot of fun running around, exercising it, too (since poodles of all sizes are athletic by nature).
A word of caution, you might want to steer clear of terriers. They can be really high-energy, and I'm not entirely certain that'd be the best for you guys (and would result in a very bored dog on flights).