Answer: Earthquake weather?

The short answer: 

Wednesday, April 18--unseasonably warm (around 80F according to some reports) and clear, but windy.  (But other reports say it only got up to 62.) 

Thursday, April 19--same as the day before, unseasonably warm and clear, but windy. No temperature data.

Friday, April 20--the usual northwesterly winds returned, cooling off the city and returning to normal San Francisco weather. No temperature data.

Saturday, April 21--temperature data lost. (But probably warm and clear.)

Sunday, April 22-- warm; high: high of 72F, no low temperature data.

Monday, April 23--0.51 inches of rain; high of 60F (SF Chronicle, April 24, 1906; NOAA records)

Tuesday, April 24- 0.14 inches of rain; high of 60F (NOAA records) 

While some of our readers found this pretty straightforward, I actually spent around 2 hours trying to get the answer.  This was difficult for many reasons.  First, it's San Francisco--land of many micro-climates (the weather can vary 20 degrees F across the city; it can be raining in the western part of the city, but sunny in the eastern.  Second, because of the disaster, local newspapers stopped publishing and the weather stations in San Francisco were destroyed. (And, as noted, with such wildly varying microclimates, you can't just extrapolate the weather nearby, in Oakland for example, to what was going on in San Francisco.)  Third, there was general chaos in the city, and noting down the weather was not first among the priorities when there were so many homeless and the city was burning.  

This all made for a great search challenge as it quickly became clear that the normal tricks-of-the-trade (such as reading archival newspapers) wasn't going to work as easily.  

I began my search with the obvious [ San Francisco weather 1906 ] and ended up reading lots of commentary that told me about April 18th and 19th.  Those days were unusually warm and clear, which meant that the immediate aftermath for the survivors wasn't miserable, their town might be burning down, but people could camp in Golden Gate Park without the cold and misery of fog.  

But getting the next couple of days data was harder.  

One of the best resources I found (after much reading through search results) was the book The Great Earthquake and Firestorms of 1906: How San Francisco Nearly Destroyed Itself.  If you search within this book for "weather" you'll find a good piece of the book describing the weather of the week on pages 51 and 52: 

p. 51.  “The weather that early morning was unusually “clear and pleasant,” according to the weather bureau.  The usual damp early morning fog was missing.  There had been no measureable rainfall for more than two weeks.  A ridge of high pressure extended over eastern Washington and Idaho with a low pressure are ot the south, forcing desert winds to flow from the northeast toward the central California coast—the reverse of the usual pattern of cool northwesterlies that blow from the ocean toward the land.”

P 52.  “The the first three days after the earthquake, the city was buffeted by wildly gyrating winds generated by the changing patterns of weather and the internal workings of the firestorm.  On the first day, the prevailing northwesterly wind and the unusual easterlies dueled for dominance, speaking the flames to  the east and the west.  The easterlies increased in tempo on the second day; the fires were driven westward.  By late Friday, a strong northwesterly wind from off the ocean had reasserted it dominance, cooling the city and forcing the fire back toward the waterfront and across fresh combustibles.  On Saturday there was rain, which ended the fires, but discomforted the many refugees sleeping in the open or temporary shelters.”

“The temperatures were ideal for a conflagration.  On the second day the thermometer jumped a dozen degress to an unseasonable high of eighty-one, as measure in Berkeley and Oakland, since San Francisco no longer had a functioning weather bureau.  Friday was also unseasonably hot. 

That tells us the weather from April 18-21.  What about the rest of the week?  And would it be possible to be more precise?  

This gets tougher. Since many weather stations were destroyed, most of the records were lost.  Luckily, the weather service is full of remarkable and pretty persistent people.  Alexander George McAdie ( Born 4 Aug 1863, died 1 Nov 1943) was at his post in the Mills Building in the 3400 block of Clay Street and kept meticulous records. (He was also vice president of the Sierra Club from 1904-1913, and was responsible for the naming of Mount Muir.)  

His records show (see the actual scans of his log books for April 1906).  This is a fascinating document as it's the official temps and precipitation.  

 

You can find the official compilation of all this data by searching for [California "climatological" data 1906] which leads you to Climatological Data California (compiled by the California State Agricultural Society).  

But this question started with what the human experience of the weather was.  And it sounds from reports as starting out fine, but growing increasingly unpleasant with the cold, the fog and the wind returning after a few days.  There's no question that San Francisco can be a chilly, windy place with "the northernly tradewinds adding to the inhabitants misery" on April 20th.  (per NYtimes, April 21, 1906)  

Search Lessons:  A big takeaway from this challenge has been how much data can be found in online Books.  Finding the Climatological data book was a surprise, although I ended up there because I saw a reference to it somewhere in my reading.  I jotted down a note, and then searched for  [California "climatological" data 1906] (the word is in quotes to prevent synonyms from being search--I knew this was in the title of the book).  

Another technique I tried, although it was slow going, was searching for diaries and letters written at the time.  Again, there's no shortage of such material, but it's slow going to extract weather information from such texts.  People weren't especially worried about the weather until it started to rain, so they didn't write much about it.  (By the way, most of the letters are absolutely fascinating when you start to see the discrepancies between the official reports of what happened vs. individual's eye witness reports.  I'm often skeptical about eye witness testimony, but the variance between the number of dead officially reported and the number of dead bodies observed is pretty striking.  The city seems to have massively underreported the casualty rate.)  

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Additional reading:  There's really a huge amount of stuff out there about the earthquake and fire.  Many of the stories are tragic, some heroic, and some are just unbelievable.  (Some authors believe that the majority of damage to the city was caused by ill-advised attempts to create firebreaks by dynamiting barriers.  This worked, except when they blew up a distillery, which caused massive explosions and sent firebrands all over the place.)  Here are three that I found especially interesting. 

SF Chronicle article about our hero, Alexander McAdie: http://www.sfmuseum.net/press/clips13.html

SF Virtual Museum articles (they have a great wealth of diary and oral histories about the disaster).  

Letter from President Theodore Roosevelt to Congress asked for the Department of War to assist in supplying San Francisco with much needed relief.  (And, interestingly, a reply from Wm. H. Taft, who became president following Roosevelt, and a few quick notes from General Funston, for whom Fort Funston in San Francisco is named.)  

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Postscript:  The images below are from a set of stereoscopic images I found while stopping in a library in Point Arena, CA, a fairly small and rural town about 100 miles north of San Francisco.  I had an hour or so to hang out in the downtown area, so I wandered into the library and discovered a pile of old stereoscope images just sitting out on a display case.  So far as I know, these have never been published online before.  (Sorry about the relatively low-res, but these were captured with my cellphone camera, which was the only device I had available at the time!) 

 This was taken on California St in downtown.  The car might have been staged,
but the background buildings are clearly completely destroyed.  

This scene is from Fort Mason, one of the major refugee sites for survivors.
The Army supplied tents and a thriving mini-city lasted for months afterwards.  

Search on!