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> 忘れてて良いのに

I understand this as: "You should forget about it"

But what does のに do?
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question
grammar
What does のに mean at the end of this sentence?
I can suppose the following situation.

A helped B out of his (her) hard situation.

B is one of the best A's friends, and A wants to be kind without being
patronizing.

However, every time when B meets A, B shows his (her) thanks.

Then A says "忘れていいのに"

The full sentence could be "まだあなたは感謝を忘れないんですね。私は忘れていいと思っているのに（どうぞ忘れてください）。" =
"You still owe a huge debt to it、but I sincerely want have you forget it."

"忘れていいのに・・・" implys such nuans.
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answer
のに is a conjunction that corresponds to "even though ～" or "despite ～".

`(clause A)のに(clause B)` = `although A, B`.

Unlike けど, it often (but not always) is followed by something
regrettable/disappointing. [Examples on
JGram](http://www.jgram.org/pages/viewOne.php?tagE=noni).

The latter clause is often omitted, and it implies the reality is something
regrettable and contrary to the Clause A.

`～のに。` = `although ～, (something contrary)`, `A, but...`, `I wish ～`.

So 忘れてて良いのに means "You could have left it (although, in reality, you recalled
and mentioned it)".

Examples:

>   * こんなに美味しいのに。 It's really delicious (and I wonder why you don't want to
> eat it)!
>   * そんなに賢いのに。 You are such a smart boy (and I don't understand why you did
> such a silly thing)!
>   * もっと頑張れば良かったのに。 You should have worked harder.
>   * 空を飛べたらいいのに。 I wish I could fly.
>

See also:

  * [How and when to use のに(=noni) - Maggie Sensei](http://maggiesensei.com/2012/06/20/how-and-when-to-use-%E3%81%AE%E3%81%ABnoni-request-lesson/)
  * [Meaning and level of 死ねばいいのに](https://japanese.stackexchange.com/q/181/5010)
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