The most common and most likely answer to this crossword clue is the 7 letter word CESSION
I've seen this crossword clue in The Daily Telegraph
C
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More clues leading to the result CESSION
Buddy explains!

The straight part
We know that "Giving up of rights" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case giving up of rights = cession.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CESSION mean?
CESSION noun- That which is ceded. Insurance: (part of) a risk which is transferred from one actor to another.
- The giving up of rights, property etc. which one is entitled to.