All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Respected Brother, you should know that displaying women's pictures to be seen by non-Mahram men are in principle prohibited. The ruling applies to publishing the pictures on Facebook or any website. The only exception to publishing them is in case of necessity or dire need, such as in the case of official documents like passports, ID cards or the means in which identity is needed to be revealed.
We would like to tell you that we are fully aware of Facebook website and similar social networking sites that involve many evil practices. For more benefit, refer to Fataawa 106627 and 84478.
However, publishing a woman's name only is less serious, though we see it is preferable to avoid mentioning her name where there is no need for that. Nonetheless, if a woman needs to publish her name in public chatting websites and fears that mentioning her real name may cause harm to her, it is permissible for her to use a pen name in order to block the means leading to temptation.
Finally, it should be noted that nothing is equal to being on the safe side, and that keeping away from doubtful matters is required.