Facebook has pros and cons. The benefits of Facebook and the harm it may cause are connected to the reasons why one signed up for it and how one uses it.
We have discussed this website in the answer to question no. 137243; please refer to it.
We do not think it is permissible for a man to add any women who are not his mahrams to his list of friends, and especially we do not think it is permissible to correspond with them and chat with them. What is more dangerous is when you can see them. That is because this opens the door to fitnah for those who get involved in it, and the tragic consequences that result from forming relationships between men and women are too many to list and too well-known to need mentioning. The Muslim should not be deceived by the Shaytaan’s making these relationships attractive by claiming that they are for the purpose of da‘wah, exhortation, advice and helping others. If a man really is keen to call people to Islam, there are millions of his fellow males who need that, so he should hasten to add them and help them. The same might be said to sisters who want to help others: they have to do that their fellow females and leave calling and advising men to other men.
In particular, see the answer to question no. 98107, where we mentioned ways in which the Shaytaan causes daa‘iyahs to fall prey to the fitnah of women on the internet.
What we hope is that our brother will be sincere to himself and avoid all doubts and close the door to fitnah; he should refrain from adding any non-mahram women to his friends list. If he has done that previously, he should hasten to remove their names from his list (“unfriend” them). It is better for him and for them. And we ask Allah to protect the Muslims in their religious commitment and protect them against the temptation of women.
And Allah knows best.