What percentage of bike vs. car accidents at night could be avoided by good side visibility?
Summary: very few. Somewhere between 20% and 90% of crashes involve a motor vehicle approaching a cyclist side-on (guess ~50%), and somewhere between 30% and 90% occur during darkness (guess ~50%). Unfortunately 40% or more are the result of drug-impaired driving or riding. So perhaps 60% of that 50% of 50% = 15% could be avoided if the cyclist had bright side lighting. 

The wide ranges are a result of small numbers and inconsistent statistics collection. As with anything that is quite variable and happens infrequently, random variation makes it difficult to see trends.

First, modern usage is to talk about "crashes" or "incidents", partly because that allows us to talk about who was at fault.  99% of crashes are caused by people who intended to do what they did but failed to do so safely.

New Zealand publishes something close to what you seem to be after: Cyclist-Crash-facts-2012.1.pdf (pdf link) that includes this summary: 


BicycleUniverse has a useful summary with references:


  Cyclist fatalities occurred more frequently in urban areas (66%), at nonintersection locations (67%), between the hours of 5 p.m. and 9 p.m. (30%), and during the months of June, July, and August (36%). (NHTSA, 2004)


Although they list a number of conflicting statistics on intersection fatalities (also 89% and 33%).

It's rare to find that type of incident breakdown in published statistics. This page from the USA, for example, doesn't even separate fatal from non-fatal accidents when attributing fault. And they talk about cyclists breaking the law but not motorists, so (for example) if a cyclist did not have a rear reflector on their bike when a motorist T-boned them the statistics would say "cyclist breaking the law, hit by car". This is regrettably common across all analyses.


  More than 15 percent (16.6% and 60 crashes) occurred when motorists
  made careless turns. Data showed that cars are four times more likely
  to sideswipe a bicycle than vice versa.


The key points there are:


  
  Nearly 75 percent of bikers were killed between 4 a.m. and 8 p.m. Nearly 20 percent were killed between the hours of 8 p.m. and midnight.
  Alcohol-impaired riding or driving was reported in more than 40 percent of the accidents that resulted in bicyclist fatalities in 2009. In a third of the crashes, either the driver or biker had a Blood Alcohol Content (BAC) level of .08 or higher (the legal limit).
  Bicycle rider fatalities occurred more frequently in urban areas (69%), at non-intersection locations (64%), and between 5 and 9 at night (28%).
  


The first point is near-meaningless (75% of cyclists were killed during 66% of the day). But 64% not at intersections suggests that side visibility is not likely to be a major issue (at most 36% of fatalities).

There's a big collection of links to papers at the Cycling Resource Centre (Australian site, worldwide links).

Examples:


  Vehicle drivers undertaking a turning manoeuvre posed the biggest threat to cyclists who were generally travelling straight on a carriageway. Those drivers undertaking a right turn manoeuvre posed the greatest threat, particularly those turning across multiple traffic lanes and in peak hour traffic conditions. Young drivers were seen as the group of drivers most likely to be involved in these types of crashes.


There is an excellent, although complex, chart on page 27 of "Road Traffic Crashes in NSW 2011 (Australia)" (pdf link) but it covers all road users and does not break out cyclsis. I suspect what you're looking for is something like this that only covers cyclist incidents.

For non-fatal accidents, BicyclingLife compares three cities (with references) and concludes that motorists failing to give way at stop signs in the most common cause of crashes.