Is the reason the sunshine is 'extra bright' after rain due to refraction of the additional water in the air? Quite frequently after the sun comes out after rain I experience a 30 minute period where the sunshine is 'unusually bright'. Such that it makes my eyes water. 
My question is: Is the reason the sunshine is 'extra bright' after rain due to refraction of the additional water in the air?
 A: There are some reasons I can think of, and they are not mutually exclusive if true.
1) There is more moisture in the sky. This enables sunlight to scatter more, and so the sky appears to be brighter.
2) The rain is able to clear dust, pollutants, etc. which normally block sunlight. With less of these particles in the air, more light is able to come through the atmosphere.
3) The wet surfaces present after raining are more reflective, and so more light reaches your eyes.
4) If your eyes were used to the limited light during the rain, then perhaps your eyes are more sensitive to the brightness. 
In any case, I don't think this has anything to do with refraction though. If refraction properties did change, this just influences the angle at which the light appears to come from, but this would not influence how much light actually reaches your eyes.
A: It may not be direct sunlight that does this.
Any wet surfaces, particularly if the water has not yet had a chance to disperse and runoff, will likely reflect more ambient light as well as direct light at different angles than they would dry (when they'd absorb more without the water to reflect).  I would typically notice this as streets become much brighter after rain (it's both a curse and a blessing if you do amateur street photography, as I do).
Also consider that just before it rains the air is likely to be carrying more water vapor than after.  It's hard to tell from your answer but there's also the issue of cloud - when it rains typically there is cloud overhead, and even if it's a cloudy day generally rain clouds will be darker and be absorbing more light.
As someone mentioned in comments that human vision is adaptive to brightness, it's worth mentioning some timescales for that.  I'm not an expert in human vision but timescales of 10-30 minutes would not be unusual from what I've read.  So if it was a little darker due to cloud before rain and relatively brighter due to brighter reflections and/or less cloud after rain your eyes would take time to adjust.  Note that this will vary quite a lot from person to person.
