Why can’t women give the Friday sermon or lead the Friday prayer?
Before I answer this question, I would like to point out that in this regard, the Sunnah set by the Prophet (sws) is that the head of state and his representatives should deliver address and lead the Friday prayers in the various mosques of an Islamic state. However, in case of any legitimate plea on their part, some other person can address and lead the Friday prayers as their authorized representative.
The implications of this Sunnah are very clear: In Islam, mosques are meant to be the fountainhead of authority. Also, there is a complete negation of theocracy. A person whom the Muslims choose as their leader shall also lead them in worship, eliminating once and for all the division between state and religion.
After the Prophet (sws), his Companions solemnly adhered to this Sunnah in the Caliphate they established. However, in later times when, owing to their own ill-ways, the Muslim rulers could not stand face to face with the public, they themselves handed over the mosques to the ulema. This was perhaps one of the most tragic incident in our history. The result was that religion lost its grace and the state its grandeur.
Today religious scholars are forced to assume charge of the mosques because of this sorry state of affairs.
In this situation, coming to your question, I think no religious objection can be raised on a lady giving the Friday Sermon. This is more like delivering a lecture. If a learned lady wants to deliver a talk on some Islamic topic in the Friday Sermon, she should be allowed to. There may be cultural hindrances to this -- but then, if they can be overcome, I personally think that there is nothing wrong with it.
Leading the prayer congregation consisting of men whether it be of Friday or of the other five prayers however, in accordance with the established Sunnah of the Prophet (sws), is the responsibility of men. Keeping in view the norms of modesty men, it seems, are more suited for this task. Of course, if a woman leads a congregation of women only, then this cannot be objected to.