I agree with Robert that humans may not necessarily make a better choice of selecting specific eggs for raising queens than the bees themselves do - I really don't know though.  I do not, however, believe that letting a split raise their own queen is the best option for queening if you want to improve your stock. IMO, if you want to improve the quality of your stock, it is better to split your hives by whatever method you prefer, and then introduce a cell from a hive that you have selected from your own operation that has whatever characteristics you want to propagate: gentle temperament, mite tolerance, high production etc. Your splits will have a queen sooner (you're introducing a cell), and you probably will end up with a better queen.
     When raising your own queens, I believe that it is imperative to introduce new stock to maintain genetic diversity and prevent inbreeding. Also, if someone is developing a strain with characteristics that you want, why "re-invent the wheel"?  Bring in a few new hives or queens and evaluate them for a season to see if they meet your breeder criteria and are adapted to the local habitat. In our operation, I don't necessarily want every split that I make to have a queen from whatever stock the split came from. I want all the splits I make to be headed by queens from stock that I deem to be the best suited to our operation. If you can add into this a closed mating yard, you will have that much greater chance of improving the characteristics of your hives. 
    If the actual point of the original post was that it is better to allow the bees to select the egg for queen rearing themselves, grafting several cells at once and introducing them into a queenless hives allows for selection by the bees.

Mike Lewis 
Austin Manitoba, Canada where it is  -27 celcius right now!