This will be a prejudice appeal, so you only have to show that your child's need for the place outweighs the difficulties that accommodating an extra child will cause the school.
Firstly, have you checked that it makes sense that your DS didn't get a place? What does the decision letter say? What oversubscription criteria did you think you were applying under (i.e. sibling at school, distance from school, etc.) and what does the letter say? If you think that it's right that you didn't get a place, then you need to appeal on the grounds that your DS needs a place. If you really expected to get in and it doesn't seem right that he didn't (e.g. you live very close to the school) then check that out first.
The second thing to do is to find out where he is on the waiting list. School places can move drastically.
Third thing is to consider if there are any other schools that might meet his needs with places as a back up plan, while your appeal processes.
In terms of your appeal, compare both schools in every way, then make a list of all the positives of the school you want. Even things like extra curricular clubs they run, or languages they offer, etc.
Remember you're looking at why your preferred school should take your DS, not why you don't want the offered school, although you can say x offers this but y doesn't.
Don't talk about transport, etc. unless you can support a genuine need to be closer to home.