I'm wanting to build a race that has the ability to shape change after a long rest.  Each of the shapes would temporarily modify aspects of the character.  Is there a way to have the an ability provide the player with an Option to choose after a long rest?  That option would then perhaps add Flying, or another option would add claw attacks and a burrow speed?

Not in a convenient way. They would need to go back into the builder to reselect this each time.

If you're not looking at anything "shared with public", you can make the ability in race as text only and then make homebrew feats to apply what you want, or items that can be "equipped" and "removed" as necessary, this will be faster than going through the builder. After each long rest the player can then change the feat or de-equip the item and equip another to another to change the benefits.