I'd like to start off with saying the stone is an amazing gameplay feature, it's so insanely useful.
But there are some things that could be added to keep it up-to-date with the changing community and allow it to be used to the fullest:

1. Allowing lions created from the wardrobe to be scryed

The inability to use a wardrobe lion can be a little tough when it comes to research and project breeding. For example, trying to scry the pass rate of a brand new marking that hasn't been put onto any lions yet, seeing how an unreleased base can look in your lines, or to see how long it could take to breed your dream king by using wardrobe lions. Though there are so many lions in game to work with, sometimes the one you are looking for just doesn't exist and you're left stumped.

Not only that, but allowing wardrobe lions to be used could result in so much fun, like seeing how two lions with 20 rosettes interact.

2. Breeding item toggle

Allowing breeding items to be selected would be incredibly helpful for predicting pass rates and contemplating running a real litter with said items. For example, the ability to apply CRB and see how many scrys it takes to produce a foldie or a lethal.
To start with, the current format of the scrying stone is perfectly quick and simple. So to preserve that, there would have to be a "Toggle simplified view" bottom underneath the ID input. There is an abundance of empty space there so the button wouldn't elongate the page at all.
When toggled off, the selection would be stored in cookies so you don't have to open and close it with every single scry. The toggled off version would look exactly as the scrying stone looks now, just with a little yes or no button.
When toggled, there would be a drop-down similar to studding pages where you can select everything you want to use, one for mom and one for dad. The selection would also be stored in cookies so you wouldn't have to reselect everything each time. The feature itself would be monumental for Leopard tails and Ochre powders.
Since there are some mutie breeding items, naturally there would have to be some kind of mutie addition. Putting muties into the scrying stone has been suggested since day one in fact, but it would be pretty messy. Even just writing the mutation but not matching the lineart would work ok.