How can I keep my split pea soup from becoming too thick during storage?
I'm afraid there isn't a lot you can do. You could use whole peas instead of split peas. But even then there will be some thickening. You could (evidently) make a thinner soup on day 1, so that on the days after, the soup would be of a normal consistency. You could do this by just adding more water and leaving the potatoes out, if you use them.

One tip that could help is to cover the soup as soon as possible, so that you don't loose a lot of steam (=water, which would also cause more thickening).

The only other advice I can give is to make it as you like it, and add some water each day you serve it, till it reaches the wanted consistency.