Well, I am definitely glad that you are all "normal people"; you can imagine what some of the situations are like, and I am sure that the mediator will be happy to work with you.
At this stage, the main focus generally should be getting the mediator on your side. The mediator will issue a report that the court typically relies heavily on when making its decisions.
This is the sell for mediation and for court: The biggest trap that parents fall into with mediation is that they are tempted to become the "gatekeeper parent". The other parent doesn't do this, they don't do this, they don't do that... what it really boils down to is a difference in parenting styles, and that is not a basis for change. The focus should usually be on what you would like to offer the child that you are uniquely positioned to provide. For example, perhaps the child is at a tender age and is showing excessive anxiety about being away from her primary custodian; perhaps the child is older and simply identifies better with their same-sex parent... it really does not matter, but you need a theme
that tells a story of why the child is better off with you. Imagine a wheel with multiple spokes; the center of the wheel is the theme; you can talk about whatever else you want (the spokes) but all subject matter should lead back to the theme. It is sometimes helpful to actually draw out your theme wheel with spokes.
Example 1 (focus):
(weak) "dad always gives Bobby American foods; I'm tired of him eating junk."
(better) "when Bobby is in my care, I offer him cuisine from lots of different cultures and places; He is at an age where he can start to appreciate those things and it will help with his development and assimilation as he gets older." (notice no mention of dad).
Example 2 (spokes):
(weak) "dad always let's Bobby stay up too late on the weekends."
(better) "I have always been the primary custodian; Bobby is accustomed to the structure that I have developed for him over the years and part of that structure is an early bedtime so he can be refreshed in the morning. I know my son, and it would be in his best interest if I am allowed to continue that structure through a sole custody
order." (again, this is not about dad or what dad can't do; it is about what mom is uniquely positioned to do for Bobby).
Lastly, if it is not too late, get and file some declarations from people who know your child and can attest to how he has been doing on days when in your care versus in his father's care. You do not want to be the mud-slinger, but there is nothing wrong with others doing it for you.
I hope that this helps. Obviously, there is no way to condense years of training and experience down to a few paragraphs, but this will hopefully point you in the right direction. Let me know if I may be of further assistance.