How are solvents chosen in organic reactions?
There is no list per se, but there is certainly a process to determine appropriate solvents for a given reaction. For the specific reaction you mention, here are some criteria you would consider to select a solvent: 


Because this is an oxidation reaction, choose a solvent that cannot be oxidized, known as an aprotic solvent.
The solvent needs to dissolve the reagents in the reaction: hypochlorite and 4-tert-butylcyclohexanol. Hypochlorite is a polar molecule, and 4-tert-butylcyclohexanol is a little polar thanks to the alcohol group. The alcohol group will be oxidized to a ketone by the reaction, so the solvent should be able to dissolve that reasonably well. Conclusion: choose a moderately polar solvent.


So a polar aprotic solvent is needed for this reaction. There are two more considerations: cost and safety

Obviously, the cheapest solvent with the lowest potential for harm should be selected.

A list of some common polar aprotic solvents for consideration might include:


tetrahydrofuran
N,N-dimethylformamide
acetonitrile
dimethyl sulfoxide
acetone


Tetrahydrofuran could be a reasonable choice, but it forms peroxides over time, which are quite explosive.

N,N-dimethylformamide has some health concerns: GHS07 and GHS08

Acetonitrile also falls into the GHS07 category: it causes cyanide to form in the body at high concentrations.

Dimethyl sulfoxide does not have health or safety concerns.

Acetone also falls into the GHS07 category: it can cause nausea and irritation at high exposure levels.

The health concerns of acetonitrile, dimethyl sulfoxide, and acetone are largely not going to pose a hazard for a laboratory experiment. All three of these can likely be found in any organic teaching lab.

Cost comparison, from Sigma Aldrich:

Acetonitrile: 2 L = $172.50

Dimethyl sulfoxide: 2 L = $208.50

Acetone: 2 L = $94.80

Therefore, choose acetone for this reaction!