A family member is riding my mare, how do you split the prize money if they win/place in a barrelrace?  I know whomever pays the entry fee gets that back, but then how to you split the rest of the money?  

Anyway you want, as long as you have an agreement ahead of time. Entry fee goes back to who paid, then maybe split the rest??? 

that's what someone told me, split 50% after entry fee is returned to who paid it.  

I guess it depends on a lot of stuff.  

Why is the person riding the horse?  Is the person doing all of the home riding/training/practice?  Is the person hauling the horse to races?  Is the horse living with them? Is person paying a lease or not and is it a quick 1 or 2 race deal or more permanent, monthly or summer? 

In any situation I've been in on a borrowed horse, the owner let me pay the entry fees and I kept the winnings.  None of these situations were 1D or rodeo winnings and it was typically a 1 or 2 time deal.  I was more than willing to give up the winnings to the owner, but they said I won them, keep them. 

If the rider is doing all of the work, housing the horse etc, but is just not purchased, I'd say winnings go to them.  If it's a couple time deal, winnings go to them.   If you're doing all the work and they are jump riding for whatever reason without a lease, I'd say you could discuss splitting them.  If they are winning 1D or rodeos and championships, I'd say split, based on amount won and how much work either of you is putting in. 

I am leaving my new-ish DTF mare with my trainer for the remainder of 2019 and all of 2020. The goal is to get verifiable 1D finishes with her for stud fee considerations and increased foal marketabiliy. He wants a couple embryos from her and so do I.

I offered to pay for her entries with him keeping all winnings. I doubt he'll let me pay them all - he'll probably roll his first winnings forward to next entries, knowing him. I will be paying for her basic maintenance, coggins, shots, teeth, and if she needs any injections. 

I feel this is a bargain and great business. 

So much better than what I typed to begin with. ;)

All families are different.  With my family memebers I would not ask for anything as far as winning splits.

In other family situations I know of the members are required to give all or at least 25% of the winnings to the member who "owns" the horse at that time.

My friend does it to where she pays for most everything besides maybe half of the vet bills (so training, though she does that herself, tack, saddles, gas, entry fees) and whatever she wins, she keeps. If it was a jump ride kind of situation, I'd probably split 50/50 minus entry fees for whoever paid them.