If I have a system that has both a hydrogen load and an electric load that is met, Homer calculates a COE, which is my cost of electricity.  What is the price of the hydrogen produced in that system?  Is it free, with all of the system costs included in the cost of electricity, or is accounted for in some way?

HOMER's main economic outputs are the total net present cost and the total annualized cost.  They include all costs of the system.  Look at the cost breakdown table in the Simulation Results window (the one you get when you double click a system) and you'll see how the total annualized cost is the sum of all the costs associated with the system.  The NPC is calculated from the annualized cost as explained in the help file.  The objective value of the optimization is the NPC, meaning that the systems are ranked according to NPC.

The levelized COE is a handy economic metric for systems that produce only electricity.  For systems that produce thermal energy or hydrogen, the levelized COE is simply not useful.

If I have a system that has both a hydrogen load and an electric load that is met, Homer calculates a COE, which is my cost of electricity.

That's not quite true.  The COE is what you get when you divide the total annualized cost by the total electric energy served (the equation is in the help file).  So it's not the "cost of electricity" in your case because the total annualized cost includes costs associated with hydrogen production.

What is the price of the hydrogen produced in that system? Is it free, with all of the system costs included in the cost of electricity, or is accounted for in some way?

As I hope I explained in the first part of this email, HOMER accounts for all the costs of the system, including the cost of producing hydrogen.  It just doesn't try to allot the costs between the two products of the system (hydrogen and electricity).  Like I said, it's not clear to me how it would.