Why the tv-script writers thought she should drop her ring, who knows. It does sound like Nika's answer is the official story. I would like to add some things, and a few thoughts.
It does seem like this is an homage to Lord of the Rings, when Merry and Pippin are abducted by orcs, and leave a piece of jewellery behind as a way to signal their friends. GRRM is fond of homages, and his books are full of them.
Not idly do the leaves of Lorien fall. If this is the case, it is rather weak, in my opinion. Aragorn and Legolas had extraordinary abilities, and followed a given track. This was in the middle of nowhere. If she expects anyone to find it, she is optimistic on the verge of delusional. If this is an actual plot line that is going to happen, it is rather far fetched.
Another explanation might be that she would get rid of evidence of her new marriage to Hizdahr, or her old marriage to Khal Drogo. However, seeing as she was discovered because she was flying a dragon, the chance of any such ruse to work would be microscopic indeed. "Say, perhaps she is some other girl riding a dragon." No.
However... SPOILER Season 6, and/or ADWD/TWOW -- Mystery spoiler, be warned!
as the script writers are not unaware, this is when Mirri Maaz Duur's prophecy starts to come true:
"When the sun rises in the west and sets in the east," said Mirri Maz Duur. "When the seas go dry and mountains blow in the wind like leaves. When your womb quickens again, and you bear a living child. Then he [Drogo] will return, and not before."
As described here, the following things have happened in the books:
- Dorne has "risen" in Westeros, and Quentyn died in Mereen, "The Suns Son"
- The Dothraki sea -- the vast grasslands around Mereen -- have gone dry with the coming winter
- Rhaegal and Viserion broke free and destroyed pyramids (mountains) in Mereen
- Daenerys' womb quickened when she had her "moon blood" (menstruation)