Who is the head coach? They will ultimately make the decision; hopefully with input (and the subsequent reasoning) from the staff. It should be a "dictatorial democracy" as the final decision should / must rest with the head coach.

My 2 cents: Write the names down the page with three columns for each. Don't tell the kids what you are doing. After practice, as the kids come out (one at a time), give them a page with all of the available numbers and ask them for their first three picks - in order. Give them 1 minute to decide. No promises; you will ensure the numbers are given fairly. Record them so the kids can't see what other kids picked. (Another coach keeps the kids busy in the dressing room and lets them out one at a time.) This way they go into the decision 'blind' and can't influence others or think about it for long. Then you (or the head coach or the staff) figures out the numbers based on trying to accommodate as many people as possible.

If there are 'ties' after the other numbers are decided, pull the names out of a hat like you said.

I wouldn't make too big a thing out of this. Look at all sides of the equation... from the coaches (adults) side and the kids. Are there other ways to 'start totally fresh' if you allow the returning players to have their numbers? Or are their other ways if you pick the numbers like I suggested above?

Good luck - let us know what you decide to do and how it works.