I figured out how to get the common items. If you give them combinations like these, then you get the following items. If you repeat an Egg Ring combination but you already have it, you'll be given the stated item. Choose any kind of Fancy or Color egg for each slot.

10 Ginger Cookies (Base: Pristine Egg Ring)
- Fancy
- 3x Plain Color A
- 2x Plain Color B

Odd Egg (Base: Vibrant Egg Ring)
- Fancy
- 2x Color A
- 1x Color B
- 1x Color C
- 1x Color D

10 Apple Tarts (Base: Brilliant Egg Ring)
- Fancy
- 3x Color A
- 1x Color B
- 1x Color C

10 Bubble Chocolates (Base: Midnight Egg Ring)
- Fancy
- 2x Color A
- 2x Color B
- 1x Color C