There is absolutely no harm and no sin in Islam if a believing girl wishes to befriend one amongst the womenfolk, or even a boy in Islam, provided that the boy is a mehram relative of the girl like her husband, her brother, her father, her uncle, her children, her nephews, etc.
Although the Quran states that contact with the opposite sex is permissible, that does not mean that it is okay to step outside the bounds Allah s.w.t. has set for us.
But if the boy is not a mehram (all those males whom a woman cannot marry at anytime in her life whatsoever. like one’s father, brother or son etc), relative, it would not be permissible nor would it behove a woman who sincerely fears Allah and the Last Day, to have any sort of friendship with him. There is no harm if one talks or deals with them for an absolutely genuine need, but to form a casual friendship or any sort of casual relationship with non-mehram members of the opposite sex would be absolutely impermissible in Islam.
Allah Says in the Holy Quran Chapter 5 Surah Maidah verse 5:
This day are (all) things good and pure made lawful unto you. The food of the People of the Book is lawful unto you and yours is lawful unto them. (Lawful unto you in marriage) are (not only) chaste women who are believers but chaste women among the People of the Book revealed before your time when ye give them their due dowers and desire chastity, not lewdness nor secret intrigues. If anyone rejects faith fruitless is his work and in the Hereafter he will be in the ranks of those who have lost (all spiritual good).
Islam does not permit the casual friendship between a non-mehram man and woman, as this sort of friendships are the first steps which sow the seeds of destructions of morality in society, which Islam vehemently tries to protect. The biggest evidence of this degradation of morality can be seen in the societies which have encouraged and allowed these casuals non-mehram relationships to flourish!
Allah says in the Holy Quran Chapter 33 Surah Ahzaab verse 53:
If you (the noble companions of the Prophet) have to ask the wives of the Prophet for something, ask for it from behind a curtain. This is a better way for the purity of your, as well as their hearts.
The above command is for some of the most noble and best people who ever lived on this planet earth; the companions of the Prophet, and the chaste and noble wives of the Prophet of Allah (saws). Allah commands the companions of the Prophet that if they have to ask for something from even the noble and chaste wives of the Prophet (saws), they should do so from behind a veil or a curtain, because this way is better and purer for both, the companions and the wives of the Prophet (saws). No people could have controlled their emotions and feared Allah more than the noble companions and the chaste wives of the Prophet; but even to such noble souls, Allah has commanded that it is purer and better that if they need to talk to the wives of the Prophet, they should do so from behind a curtain!
In light of the Holy Quran, and the various authentic narrations of the Messenger of Allah (saws), with the exception of marriage, it is absolutely forbidden for a non-mehram woman to have any sort of casual friendship with a non-mehram man.