I loved reading the responses! Of course you can try to keep just the best ones or the ones that show progress or a new skill, but it is hard to throw things away that your kid made!
An idea I just heard of is to make Care Envelopes for their grandparents who live far away. You put in artwork, stories, (anything the kids make that you don't want to keep! They won't know they are getting your "second bests!"), and don't forget to add a few recent photos of the kids. Grandparents will LOVE getting these Care Packages every month (or whenever you do them, but once a month is a good schedule).
One mom I know frames her kids' artwork and displays it around the house. The kids are involved in choosing the mat and frame and even when to retire a certain piece and replace it with a new one. As they get older, the art gets better!
Since your child is young still, you could pin the artwork on a bulletin board and change them out each week. I like the idea of letting the child pick which ones to keep.