Yes, women are allowed to dye their hair and the recommended colour is with henna (a red plant dye).

(Fatawa Hindiyya V.6 pg 238)

The reason for prohibiting the use of black colour to dye ones hair is that one is deceiving ones age and appearance.

Shaykh Taqi Usmani mentions in his commentary on Sahih Muslim, Takmila Fath al-Mulhim, regarding women dying their hair:

As for women dying their hair in order to beautify themselves for their husband, Qatada (May Allah be pleased with him) gives permissibility as mentioned in Musannaf Abdur Razzaq 11:155. Also Ishaq gives permissibility as narrated by Ibn Qudama in his Mugni (1:76). Shaykh Taqi ends with saying, ‘I have not seen any other statement beside these two.

Allah Knows Best

(Takmila Fath Mulhim 4:150)

Dyeing the hair with henna would be more recommended as was the practice, but other colours are also permissible.

The Prophet (Allah bless him and give him peace) advised the father of Abu Bakr (May Allah be pleased with them both) to change and dye his hair. He said, ‘Change and dye your hair and refrain from using black’

(Abu Dawud 2.137)

This hadeeth is with regard to advising a man in using a dye. As for women it gives permissibility as she is more urged upon and advised to beautify herself for their husbands.