I am interesting in differences in affinities between Cu(I) and Cu(II) ions towards the same coordination environment. My main focus is related to proteins, however simpler systems might be also useful due to limited data for some proteins.
There are many factors that govern the process but generally It depends of what groups are going to bind to the metal nucleus. Using hardness sofness criteria we can predict what ligand can form a complex with the bigger stability constant (strongest bound). For example pyridine is a hard ligand so Cu(II)-Pyridine bound will be strongest that Cu(I)-Pyridine ( Cu(I) is softer tan Cu(II)). NOTICE THAT THIS IS NOT A STRICT RULE, there are many factos to take account. An example of this isthat ligands with cysteine groups (the thiol group) do a redox reaction with Cu(II) forming Cu(III)
Cu(II) complexes are generally more stable than Cu(I) complexes. To reverse this order, special conditions have to be achieved. In proteins this is done by fixing the geometry to a shape less favored by Cu(II) (distorted tetrahedral) and also by ligating copper with soft ligands such as cysteine.
The following lines can be useful for arriving at the answer of the posted question:
There can be a generalization that copper (I) as a d10 system with completely filled orbitals should be more stable than copper(II) with d9 configuration which do not have completely filled orbitals. However this is not always so, this brings first correlation that stability is not absolute but relative, as copper(I) in aqueous solution disproportionates to copper(II) and metallic copper(0) meaning when we consider aqueous solution, copper(I) is unusual oxidation state and unstable relative to copper(II) which is stable and usual oxidation state in aqueous solution. While as in a non-aqueous solvent like Acetonitrile copper(I) is way more stable to copper(II).
The stability of an oxidation state in a coordination compound depends on its hard soft nature. Copper(I) is a soft metal ion and hence prefers soft donor sites like sulfur and Iodide and not so much preference for borderline or harder donor sites. e.g the most common complexes of copper(I) are with sulfur donor ligands like thiourea. The soft -soft interaction of copper(I) and thiourea is so dominating that when thiourea is added to the solution of copper(II) it oxidizes thiourea to reduce copper(II) to copper(I) (see attached ic PDF and journal of Analytical chemistry). Copper(II) is borderline hence forms more stable complexes with borderline donor like nitrogen e.g the famous dark blue [Cu(NH3)4]2+ complex.
The stability of an oxidation state also depends on the preferred geometry. Copper(I) being d10 system can SP3 hybridize and hence prefer tetrahedral geometry while copper(II) being d9 system prefers distorted octahedral geometry on account of Jahn Teller Stabilization energy. Hence Copper(I) in tetrahedral systems is more stable than copper(II) while as copper(II) in distorted octahedral systems shall be way more stable than copper(I). This can illustrated from the binding constants of copper(II) and copper(I) with neocuproine ligand (see attached pdf 2 which highlights how complexation changes the stability of copper(II)/copper(I) and influences the redox potential of Cu2+/Cu+ redox couple)
The stability of an oxidation state in a given complex depends on the type of bonding. Synergistic bonding (ligand donation and back acceptance) brings in electrical neutrality in complex which brings additional stability in the complex. Copper(I) on account of its lower charge and completely filled d orbitals favors a good degree of back acceptance to ligand compared to copper(II) hence copper(I) forms more stable complexes with pi acceptor type ligands while as the stability of copper(II) complexes with pi acceptor ligands is less as compared to copper (I) due to low degree of back donation from copper(II).
copper(II) is a typical case of Jahn Teller effect and hence has a stability in distorted octahedral geometry where it enjoys Jahn Teller Stabilization e.g in complexation of copper(II) with ethylenediamine(en) in aqueous solvent only two en ligands get attached to copper while as the binding of third en is restricted as it will make the copper to lose the distorted octahedral configuration and hence Jahn Teller stabilization.(see Inorganic chemistry by James Huheey 4th edn page:454-455) No such distortion/ stabilization is seen in case of copper(I)
In summary we cannot generalize whether copper(II) is more stable to copper(I) or copper(I) is more stable to copper(II). It can only be decided as per the system under consideration.
Post note:
Copper(II) and copper (I) have same atomic number same nucleus but just one difference of electron which makes the huge difference in the stability of copper(II) and copper(I) compounds make one to appreciate the diverse and beautiful chemistry of transition metal systems which has little or no parallel in carbon or main group chemistry.