So, let's say we have a sublist-from-a-saved-search on a record....would it possible to be able to have a button next to each entry in the sublist with a button that calls a script based on a value in that row of the sublist?

You can't have buttons next to each line with the native netsuite apis. You could probably just trigger whatever you want to happen with a checkbox or something, would probably make more sense to check each line then click the button and read from the lines that are checked.

Thanks. What I am looking to do is on each row, there's a value of an ID of a file stored in another system, and the hypothetical button would pull that file out of the other system through an API.

Have you tried using links instead of buttons in the columns of the search ? With additional formulas it should be okay to fetch the any current value, and make it custom link based on that