As far as recycling or rejuvenating HCl from a copper chloride solution would be very difficult from a simple electrolytic cell setup.
To setup for this discussion I will use some formulas.
Copper being a fairly un-reactive metal, will not dissolve in a non oxidizing acid like HCl, being placed below hydrogen in the reactivity series of metal.
Cu + HCl --> No Reaction.
Copper metal need to be oxidized (an electron removed) to dissolve in HCl, and to form chloride of copper compound dissolved into solution.
Air, oxygen in water, H2O2 hydrogen peroxide or some other oxidizing agent to oxidize the copper to copper oxide, so the copper oxide can react with the HCl to form copper chlorides dissolved in this water solution.
CuO(s) + 2H+ -->Cu2+(aq) + H2O(l)
CuO + 2HCl --> CuCl2 + H2O
Cu + 2HCl + H2O2 --> CuCl2 + 2H2O
Notice in the reactions above the hydrogen from the acid, the oxygen from the air, H2O2, or from the oxygen of the copper oxide form water in solution.
2H+ + O 2- --> H2O (aq)
HCl, hydrochloric acid, hydrogen and chlorine gas in a chemical redox reaction forms the acid.
H2+ + Cl2- --> 2HCl
Basically the Hydrogen is what makes an acid an acid, and this hydrogen or acid is used up in the reaction, to make the more stable water (although we can have some excess acid in solution depending on our solutions), the hydrogen that forms the chloride acid (hydrochloric acid, HCl) is used, reacting with oxygen or oxides to form water.
When we dissolve the metals in acid or oxidizers we are transferring electrons, like from copper atoms we oxidize (remove an electron from the copper metal atom), The copper is now an ion (Cation), the oxidizer or acid is reduced (gain of electron) to an ion (anion), in this case the Cation Hydrogen (from the acid) and the anion from the oxide or oxygen form the compound of water, the copper forms an Cation with the anion of chloride (reduced acid HCl) to form the compound of copper chloride in this solution of water, the chloric of the acid being reduced to chlorides with the copper as copper chloride salt dissolved in water.
Now when we use electrolysis we can separate the compounds in solution, the compounds of copper chloride can be separated into the anions and Cations in the cell, the Cu+ Cation moves to the cathode compartment, or side of the cell can gain an electron (from our external power source or battery) and plate out as copper metal, this ion being reduced back to metal, as the copper ion gets back the missing electron.
And the chloride anion moves towards the anode where it loses an electron, being oxidized back to elemental chlorine and leaves the solution as gas or can stay in solution (depending on conditions) note it would be harder to keep this solution in the liquid under the conditions most simple cells are run under.
What about the Hydrogen from our original acid that made water? well we can also split this water compound of water in our electrolytic cell , forming hydrogen gas at the anode, and hydroxide at the cathode...
but getting the hydrogen and chlorine gases to form back to HCl acid would not easily work in a simple cell, the conditions these gases form at in the cell may not be under the same conditions of cell operation, capturing of these gases to remake our HCl acid would not be an easy task...
It would take a lot more explanation to give more of an explanation as to why this would not work easily to re-make HCl acid, besides HCl is so common and cheap it would not be worth the trouble, it would be easier to make HCl from sulfuric acid and salt than to try and get it out of copper chloride with electrolysis, at least on a small scale in a homemade electrolytic cell.
Although copper metal is a fairly un-reactive metal, it will oxidize easily with air or oxygen at high heat like when trying to melt it, forming copper oxides which would form a slag of copper oxides, instead of reducing the copper to metal in the melt, for melting copper we need to remove oxygen from the copper and the melting environment, to do this we can use chemistry in our melt, using a reducing flame to heat the environment, use a charcoal fire limiting oxygen in the melt, and or using a carbon source in our flux like flour, charcoal, sugar, sodium carbonate..., to remove oxides or oxygen from the copper and its environment in the chemistry of the melt.
There are other times where we may wish to oxidize the copper in a melt, using an oxygen rich environment or a flux like a nitrate to oxidize the copper into an oxide so it will report into the slag.
Melting metals is a chemical reaction where metals can be oxidized or reduced (again transfer of electrons), (just like other chemical reactions we perform with metals), depending on the metal, and how it reacts with oxygen or other chemicals in the melt, we can change the chemical properties of our melt to get a desired chemical reaction, in the melting environment or with flux or both.
I do not know how well I explained this, or if it will help any with your questions or not.
A few mistake corrected Thanks Dave