First of all, let it be said that it is not possible to convert every single cat into a "cuddly lap kitty," though there is no harm in trying. It would be difficult, if not impossible, for example, to take a formerly feral cat and convert her into a feline lap-lover that was fawning on anyone's lap. Experiments in England have shown that if cats are raised without human company for the first 7 weeks of their lives, they will never be fully accepting of people. The best you could expect from a cat with this kind of background is occasional fleeting visits during which the cat might tolerate a modicum of petting. This level of trust on the part of a cat like this represents something of a psychological breakthrough.
Another reason why some cats do not take well to the job of being lap cats is to do with inherited disposition. Some cats, by nature, are more independent and aloof than others; whereas some are just plain fearful. Such traits manifest as an anti-social nature with respect to would-be human companions. Some of these reclusive cats may be coaxed out of their shell by kind and patient treatment, but even the best results that can be achieved in terms of friendliness to people may be a far cry from relaxed lap sitting.
You should recognize these "exceptions to the rule" before trying to convert all comers to the noble art of lap sitting and the acceptance of liberal petting and cuddling. Nevertheless, the majority of cats are trainable this way as long as the owner goes about the process in the right way. There are some general rules that owners may want to consider when trying to forge such a close relationship with a cat.
The Way Forward Where possible, select a cat that is the product of affectionate parents.
Obtain a very young cat - it's almost a case of the younger the better (though kittens adopted when they are too young can present the opposite problem of over-bonding or over-attachment).
Raise kittens with kindness and never physically punish them or yell at them.
If it is too late for any or all of the above, and the cat is already somewhat wary or reclusive, it is never too late to start trying to repair existing damage.
The general philosophy for successful rehabilitation is to create circumstances favorable for the cat to approach the owner, rather than the other way around. Striding up to a cat, thus invading her flight distance, apprehending her and placing her on your lap, thus invading her personal space, is exactly the wrong approach.
Arrange for rehabilitation to occur in quiet circumstances. Position yourself in a large room with the cat, and arm yourself with a good book and a bag of food treats that your cat finds delicious. The procedure will go more swiftly if you arrange for the cat to be slightly hungry at the beginning of the session as this will increase the cat's motivation to accept the food treats.
Without moving from your comfortable chair or couch, toss a food treat in your cat's direction and be patient, until she finds and consumes it. Repeat this procedure at intervals, dropping the food progressively closer to yourself and, finally, beside yourself on the couch or chair.
Next, arrange for the cat to take a food treat from your hand, gradually moving your hand toward your lap, only releasing the food treat if the cat puts her paws up on your lap.
Remember that you will certainly not be able to make a reclusive cat into a cuddly lap-sitting cat in one session. The whole process may take several weeks or even as much as a year. Be patient and be grateful for modest improvements. Never attempt hurry things along; never come on too strong; and never try to force the issue. Allow your cat to be drawn into a vacuum of food, affection, and petting that you provide for her.
Sometimes you can focus a cat on what you are doing more acutely by employing a "click" to signal the delivery of a treat. This focuses the cat's attention on you, the source of the click, and cues her to the subsequent gift of the food treat from you, i.e. you become the common link. The use of a clicker in this way may help quicken the retraining process. Clicker trained cats seem to have more interest and faith in their owners than untrained cats.
The person trying to build the relationship with the cat should be the one to feed her regular food. It helps to have the cat 'meal fed' and to have the meals put down as obviously as possible by the person wishing to forge the close bond.
The person trying to draw the reclusive cat out should probably arrange to play games with the cat at least a couple of times a day. Moving toys are best, such as cat dancers and pull toys on a string.
If an appropriate combination of such measures is engaged in by a well-meaning cat owner, there is no reason that, over time, a relatively reclusive cat shouldn't be encouraged to come forward and interact affectionately. In many cases, lap sitting will then occur spontaneously, with its implicit permission to pet and cuddle. One caveat, however, is that if the cat wants to escape from the situation, or has had enough for any reason, she should not be restrained but should be allowed to hop off your lap at her pleasure. Cats are at their best when they are allowed to come and go as they please.
In many cases, all it takes to produce the ideal, easily pet-able lap cat is to arrange for all the good things in life to come only and obviously from you. As Konrad Lorenz so aptly put it with respect to training, "art and science aren't enough, patience is the basic stuff." And, you may have to be patient for quite some time. I have one cat that was skittish from the time that I rescued her and she only became a completely cuddly lap cat at the age of 12, after years of catering to her and two geographical moves. Actually, I think it was one of these moves into a small temporary lodging that confined her close enough to my family that she had no alternative but to interact with us.
The moral here may be that although you don't want to force your presence on a cat, you also don't want to provide the cat an opportunity to always be so far away from you that she never has to interact with you. And, for those few cats who never come round to becoming fully conversant with, or accepting of, lap sitting or cuddling, remember that this apparent shortfall does not necessarily mean that they have no affection for you, the owner. It may simply be that they show their affection in other ways.