Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
According to some scholars, it is permissible for a woman in her waiting period to stay wherever she wishes, and it is not prohibited for her to make hajj or Umrah during her waiting period.
Answering your question, Dar Al-Ifta Al-Misriyyah, states:
It is established in Islamic law that a wife whose husband passes away must observe a waiting period of four months and ten days.
God Almighty says: {And those who are taken in death among you and leave wives behind – they, [the wives, shall] wait four months and ten [days]. And when they have fulfilled their term, then there is no blame upon you for what they do with themselves in an acceptable manner. And God is [fully] acquainted with what you do.} (Al-Baqarah 2: 234)
Al-Furai`a bint Malik ibn Sinan, Abu Sa’id al-Khudri’s sister (may God be pleased with them), said that the Prophet (peace be upon him) ordered her to remain in her house until the end of her waiting period. This was recorded by Imam Malik in his al-Muwatta`, al-Shafi`i, Ahmad, Abu Dawud, at-Tirmidhi, an-Nasa`i, and Ibn Majah. Ibn Habban and al-Hakim declared it authentic.
The predecessors differed on whether remaining in the home is among the requirements of the waiting period. The majority of jurists from among the Predecessors and Successors maintained that a woman in her waiting period must remain in her house and it is impermissible for her to go out for hajj or for any other reason basing their opinion on the above hadith.
However, it was reported through some of the Companions and their students that it is not obligatory for a woman to remain in her home upon the death of her husband. According to them, it is permissible for a woman in her waiting period to stay wherever she wishes, and it is not prohibited for her to make hajj or Umrah during her waiting period.
This opinion was reported from Ali ibn Abi Talib, Abdullah ibn Abbas, Aishah and Jabir ibn Abdullah (may God be pleased with them) from among the Companions.
It is also the opinion of Hassan al-Basri, Jabir ibn Zayd, and ‘Ata` ibn Abu Rabah, as well as the Zahirite scholars.
They based their opinion on the fact that the verse legislates the obligatory nature of the waiting period yet does not stipulate that a woman remain in her house. Unlike the majority of scholars, they contended that the above hadith is weak and, even if it were sound, it applied to a specific incident and cannot be generalized.
Though the opinion of the majority of scholars is preponderant because it is based on a sound hadith, it is conditioned on the legal principles: “Objections are not for issues where there is a difference of opinion, but in matters where there is an agreement,” and “It is permissible to follow the opinion of those who maintain permissibility when there is a need.”
It is permissible to follow the opinion of those from among the Predecessors on the matter relating to obligatory hajj, especially for a woman who may not have another opportunity or one who has already paid the exorbitant expenses of hajj.
Almighty Allah knows best.
Source: http://www.dar-alifta.org/Foreign/ViewFatwa.aspx?ID=1340&text=hajj