Can women gather in the house of one of their number to offer Taraweeh prayer?

Firstly: 

It is permissible for women to gather to pray Taraweeh in the
house of one of them, subject to the condition that there is no wanton
display of adornments (tabarruj) involved in their going out, and that there
is no danger of fitnah. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

There is nothing wrong with women attending Taraweeh prayers
so long as there is no danger of fitnah, subject to the condition that they
go out in a decorous manner, not making a wanton display of their adornments
or wearing perfume. End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen,
14, question no. 808. 

But it is better for
each of them to pray on her own in her house, and in an inner part of the
house. The Prophet (peace and blessings of Allaah be upon him) said
that it is better for women to offer obligatory prayers in their houses than
to pray in the mosque, so it is more apt that this should also apply to
naafil prayers. 

It was narrated from Umm Salamah (may Allaah be pleased with
her) that the Prophet (peace and blessings of Allaah be upon him) said:
“The best mosques for women are the innermost parts of their houses.”
Narrated by Ahmad, 26002; classed as hasan by al-Albaani in Saheeh
al-Targheeb, 341. 

Indeed it is better for a woman to pray in her home than to
pray in congregation in al-Masjid al-Haraam (in Makkah) or in the Prophet’s
Mosque behind the Prophet himself (peace and blessings of Allaah be
upon him). 

It was narrated from Umm Humayd, the wife of Abu Humayd
al-Saa’idi (may Allaah be pleased with them both) that she came to the
Prophet (peace and blessings of Allaah be upon him) and said: “O
Messenger of Allaah, I love to pray with you.” He said: “I know that you
love to pray with me, but your praying in your room is better for you than
your praying in your house, and your praying in your house is better for you
than your praying in your courtyard, and your praying in your courtyard is
better for you than your praying in the mosque of your people, and your
praying in the mosque of your people is better for you than your praying in
my mosque.” So she gave orders that a ‘mosque’ be built for her in the
innermost and darkest part of her house, and she used to pray there until
she met Allaah (i.e., died). 

Narrated by Ahmad, 26550; classed as saheeh by Ibn Khuzaymah,
1689; classed as hasan by al-Albaani in Saheeh al-Targheeb, 340. 

Imam Ibn Khuzaymah included this hadeeth in a chapter
entitled: It is preferable for a woman to pray in her house rather than in
her courtyard, and in the mosque of her people rather than in the Mosque of
the Prophet (peace and blessings of Allaah be upon him), even though
one prayer offered in the Mosque of the Prophet (peace and blessings of
Allaah be upon him) is equivalent to one thousand prayers offered in any
other mosque. This is the evidence that the words of the Prophet (peace
and blessings of Allaah be upon him), “One prayer in this mosque of mine is
better than a thousand prayers in any other mosque” refer only to men and
not women.  

Al-Shaykh ‘Abd al-‘Azeem Abaadi (may Allaah have mercy on
him)said: 

It is better for women
to pray in their houses because then there is no danger of fitnah. This
ruling is even more emphatic because of the wanton display (tabarruj) and
adornment of women.  

‘Awn al-Ma’bood, 2/193 

Secondly: 

If women gather together in one house in accordance with the
conditions mentioned above, it is permissible for them to pray in
congregation. The one who is leading them in prayer should stand in the
middle of the (first) row, not out in front, and she should not lead men in
prayer even if they are her mahrams. She should recite our loud just as men
do in the prayers where reciting out loud is required, so long as no men
except her mahrams can hear her voice. 

It was narrated that Umm
Waraqah bint ‘Abd-Allaah ibn Nawfal al-Ansaariyyah asked the Prophet
(peace and blessings of Allaah be upon him) for permission to appoint a
muezzin in her house who would call her to prayer, and he told her to lead
the people of her household in prayer. 

Narrated by Abu Dawood, 591; classed as hasan by Shaykh
al-Albaani in Irwa’ al-Ghaleel, 493. 

And it was narrated from ‘Aa’ishah that she would give the
call to prayer and lead other women in prayer, standing in the middle of the
row. 

And ‘Aa’ishah led women in obligatory prayers, standing in
the midst of them. 

And it was narrated that Hujayrah bint Haseen said: Umm
Salamaah led us in prayer standing in the midst of the women. 

And it was narrated from Umm al-Hasan that she saw Umm
Salamah, the wife of the Prophet (peace and blessings of Allaah be upon
him), leading women in prayer, standing with them in their row. 

Shaykh al-Albaani (may Allaah have mercy on him) said, after
narrating these reports: 

In conclusion, these reports are sound and may be acted upon,
especially since they confirm the general meaning of the words of the
Prophet (peace and blessings of Allaah be upon him): “Women are the
twin halves of men…” 

Sifat Salaat al-Nabi (peace and blessings of Allaah
be upon him), p. 153-155. 

Ibn Qudaamah (may Allaah have mercy on him) said: 

She should recite out
loud in prayers where it is required to recite out loud, but if there are
any men present, she should not recite out loud, unless they are her
mahrams, in which case she may do so. 

Al-Mughni, 2/17 

And Allaah knows best.