How are solvents chosen in organic reactions? Just reading through organic chemistry book (Modern Projects in Organic Chemistry by Mohrig, et al.), and it seems like the choice of solvent is usually arbitrary. 

One of the experiments designed uses hypochlorite ion as the oxidizer and acetone as a solvent to oxidize 4-tert-butylcyclohexanol, but it isn't clear why this solvent is chosen.

I'm curious if there is a list that gives the suggested solvents for different kinds of reactions, as well as some insight for this particular experiment.
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!