What are the rights upon a woman who seeks to separate from her husband by paying back a portion of her dowry (known as al-khul'a)?

What is khul'a
In the Shari'ah, khul'a means the cancellation of the marriage contract in return for a remuneration given to the husband, and effected by the pronouncement of the statement for khul'a. All scholars — past and present — maintain that this kind of separation is legally permissible and base this permissibility on the following verse:
"It is not lawful for you to take back anything that you have given (your wives), except where both fear that they cannot maintain (the marriage) within the bounds set by Allah: if you [arbiters] suspect that the couple may not be able to do this, then there is no blame on either of them if the woman opts to give something for her release (from the marriage contract)" (Quran 2:229).
In addition, the permissibility for khul'a is based on the hadith narrated by Ibn 'Abbas (may God be pleased with them both) who said that Thabit Ibn Qays' wife approached the Prophet (peace and blessings be upon him) and said, "O Messenger of God! I do not find blame with Thabit Ibn Qays in anything concerning his character or faith, but I dislike to disobey the teachings of Islam." The Prophet (peace and blessings be upon him) said, "Do you agree to give him back the garden he gave you as mahr (dowry)?" She said, "Yes." The Prophet (peace and blessings be upon him) then told Thabit, "O Thabit! Accept the garden, and divorce her" (Bukhari).

Islam respects customs and culture
It has been established that customs which do not contradict the venerable Shari'ah are among the integrals of the Islamic legislation. This is based on the statement of Ibn Mas'ud (may God be pleased with them both) who said, "What Muslims deem to be good is good in the eyes of God, and what Muslims deem to be offensive, is offensive in the eyes of God" (Ahmed).
The jewelry customarily presented to the bride in certain countries (such as Egypt, where it is known as shabka) is usually considered part of the dowry.

The ruling
The wife who wishes to be separated from her husband in return for a remuneration must return to him the dowry he gave her (this includes the shabka, the marriage payment received in advance, and the wife's belongings which he had given her). She must also waive her future right to financial support during the 'idda (waiting period), and any amenity payment which she might have been entitled.
God, the Almighty, knows best.