Here is a common sense approach to setting the minimum wage.
The federal minimum wage should be set to protect workers when the economy is very poor and people are desperate and can be taken advantage of.
The amount to set the minimum wage is what a consensus of companies must pay to attract workers when the economy is good (or overinflated like now).
What I have seen in some of the lower income states are around $10/hr right now.
That is what I would set the federal minimum wage at. Higher income states could set their rate higher but no state could be lower.
The main argument is that an increase would cause higher unemployment and higher prices. Setting the minimum wage when companies are already paying that wage and still can't hire enough people will do neither.
Historically, when minimum wages have been set, the amount adjusted for inflation was very close to that $10/hour.