First, you should measure the actual current draw of the router at full load (try transferring a large file). Does it actually use 9v 1A, or is that the rated power supply it came with? You might find that it uses less than 1A. You could also open the router and check to see if it regulates the 9v down to something else. If it uses a common LDO regulator, you might be able to use a lower power supply, or you might be able to replace the regulator with a better one, or remove the regulator and use a supply that matches the regulated power.
In my experience, a Fonera pocket router uses a 5v 2A power supply, but is regulated down to 3.3v with an 1A LDO with a 1.2V drop out, very inefficient. The newer version of that same router uses a switching regulator instead, which solved much of the heat issues. I replaced the regulator in mine with a switching regulator, allowing for efficient battery power operation with little heat issues.
Once you have figured out the minimum voltage you need, and the minimum current you need, you can choose an appropriate sized solar panel (or panels). It will most likely be less than the (9v * 1A) 9 Watts that you originally needed. But keep in mind that solar panel ratings are for full unobstructed sunlight. So a bigger panel would still be "better", despite the size and cost.
The best option is using the solar panel of a decent size, to charge a battery setup, and then run the router off the battery. This way, the amount of sunlight needed would be independent of the actual sunlight available when you need to use the router. And it would allow you to use the router at night.