An item's "worth" to a shrine is generally its economic value. There isn't a minimum. The green sliding bar is how close you are to earning an emblem.

Reasons to donate are:
earn mood points (get your mood up)
earn imagination points
work toward an emblem
earn favor points that you can spend to increase speed on learning a skill, if you want (spending will zero out the green bar).

You may occasionally find that donating something causes the shrine to say "feh" and give you no points, which means you donated something that has little to no economic value, like one cherry, for example. But it's fun to try donating different things and see what happens.

Using Fairly or Extremely Hallowed Shrine Powder will increase the value of your donation.

:)