What battery and solar panel voltage for Arduino Uno weather station?
I have done something very similar in the past. I used the UNO bootloader on a breadboard design.

I used the jeelib sleepy to power down my chip in between data gathering cycles (30 seconds in my case).

Instead of batteries I eventually settled on x2 2.7V recycled 100F pseudocapacitors and some cheap solar cells from ebay. I stored the data to an SD shield that I also bought from ebay.

The whole setup cost around $20 and has been in operation for over a year.

If you make sure your input voltage is not too high (The datasheet says 5.5v max but mine get up to 6v at times) and your oscillator low enough the whole setup ends up using only a few milliamps on average and will have a wide voltage range. Mine usually dies at around 2.7V clocked a little under 4MHz. The solar cells are wired in sets directly to the capacitors with a blocking diode to slow self discharge during the night. 

My design was a little short sighted and if I ever redesign it I will go with smaller solar cells, I did not realise just how efficient the Atmega328p chips actually were so the capacitors reached full charge very quickly and end up over their rating for most of the day. The biggest user of power ended up being the self discharge of my capacitors.

I did some early testing with x2 5.5V 1F capacitors and they were able to get the thing running 24/7 (not running the SD module) in Tasmania. I settled on pseudocapacitors due to their large number of cycles, their cheap price when bought from ebay recycled and the ability to run the SD card module. My original test design used x2 3V 1W solar panels in series from ebay. I did not test their actual power other than in the actual running of the device. When my pseudocapacitors arrived I rewired them into the configuration stated above, put it all in a sistema cookware box and positioned it for full solar coverage from the morning sun. To access the data I simply needed to copy the text file from the SD card to my laptop and then connect+disconnect the reset pin on the barebones board to reset it for gathering more data (The SD library would crash when the card was taken out).