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I'd look at the condo gains this way:
The condo was $400k, and Lily + her mom paid $90k up front. That's right around 20% of how much the condo was.
Next we need to know how much Lily has paid into the mortgage before Mike came into the picture. There are complexities in here as well because mortgages are inherently interest-heavy in the beginning, and light near the end, so I don't think a linear calculation on the contribution is fair. But essentially, let's say the total 30 years' worth of mortgage payments would amount to $800k, and Lily has paid it down to $700k by herself before Mike came into the picture, and they started living together.
To me, this means Lily has paid off another 10% of the apartment, meaning that she and her mom should be entitled to at least 30% of the apartment's worth.
Now, the apartment sold for $749k. 30% of that is $224.7k. That amount needs to go to Lily and her mom. How they split that between themselves is their problem.
This means there is $524.3k of proceeds from the condo sale. Assuming you do a 1/2 and 1/2 split here, each person gets $262.15k.
The re-financing of $521k needs to be looked at on its own as a completely separate transaction, and I am not sure where it came from. Did Lily and Mike buy a new house together? As they are a common law couple now, they'll need to work out their own split on how much each person pays into that $521k mortgage.
They also need to ask themselves how much from that $262.15k each person wants to contribute back into their new mortgage.
There is no "fairness" question to be asked. The only fair and definitive part is the intial 30% (or however much it was) that Lily and Mom and contributed, and ROI needs to be calculated for that portion. You can't just pay Lily $70k, and Mom $20k, and call it a day.
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