Hi C.,
As a mother of three daughters (now ages 25, 23 and 21) and also a retired teacher, I have traveled down your paper path!! I used to save a LOT of their work-especially elementary things. As they got older, they sort of weeded some of it on their own. To save everyone's sanity, here is what I would propose.
Save your child's papers for one month. At the end of the month sit down with your child and go through all of the papers and decide which one is the Best. Establish some sort of rubric or criteria that is clear to you and your child what is worthy of saving. Then save that paper. File it in a place that your child will know is safe and that he has access to-like a file cabintet. At the end of the school year, decide which of the nine is the best and save that one. Learn (and teach your child) how to distinguish if the paper is just a practice paper or if it is a pretty good one (a test, or super comments that the teacher put on the paper.) Keep in mind that as the year progresses, your child will be improving. So if there is a "tie" for one particular month, save both, but know that by the end of the school year that there should be some awesome "saves".
It gets more difficult as there are "projects" and models, dioramas, etc. that take up space, collect dust and can be cumbersome to store. After storing them for the duration of the school year and before discarding them, take a picture of the project.
Then, during the summer, you and your child could put together a scrap book of some of his featured BEST work for first grade. The papers would fit because they are nice and flat, and the photos of the projects would be a good remembrance of first grade. Any other certificates, etc for that year would also be included in a neat "School Days" scrapbook.
As the years go by, your child will become more adept at self selecting work that he knows is his best.(Kind of like his own personal portfolio).
A month might be too long of a time to have a first grader to wait, so perhaps do it at the end of the week to stay on top of it. I know that "life" will get in the way to be so organized and regimented about this, but even if you do it periodically, it will help. Rainy days and snow days (and summer vacation) will be a good time to take a few minutes to save special work.
These are just a few suggestions, but perhaps you can tweak it and have it work for you. Everyone is different, so try it and see how it will evolve. Believe me, some of the things that I have saved from my daughters and have shown them later, they couldn't even remember that they did the work. (Of course they were a lot older,too) This will help to lighten your load and involve your child at knowing what they have done was their best. It is also a good tool to use and reflect on to see their own growth as they pass through each of the grades.
Good luck.
J.