There are two ways I can answer the question as to whether or not women should have rights: Yes and No. Let's look at "no" first.
I know we have some very astute readers who will point out to me a valid reason that women should not have rights, and so I know I cannot answer the visitor's question without first addressing that perspective here.
Our insightful readers will naturally point out that the Bible says: "The earth is the Lord's, and all it contains, the world, and all who dwell in it." (Psalm 24:1) Therefore, women should not have rights because no one, Christian or non-Christian, has any rights to begin with! (If women had rights and men didn't, that wouldn't be fair, would it?) Our money, our possessions, our talents, even the very air we breathe does not belong to us. It is borrowed and belongs to Someone else. Technically, women should not have rights (nor should men) because everything we see, hear, touch, taste, and smell belongs to God. (Psalm 24:1 states that our very selves are God's possessions.) If we have anything, including our "rights," it is for one reason and one reason alone: He has graciously chosen to give that to us.
The Bible further states that when a person becomes a Christian, that person willingly gives up all those things they perceive to be their "rights". We become slaves to Christ. We do what our Master tells us to do. Fortunately in our case, we serve a perfect Master, who will never steer us down the wrong path. We can trust Him with whatever instructions He gives. That being said, I can now move on to the response that I'm sure our visitor was looking for.
Throughout this site from time to time, we have featured some articles marking our feelings regarding the ideology known as "feminism." Generally speaking, we at Reformed SHEology do not adhere to feminist values, and at times have openly criticized feminist ideology on this blog. Why? Because it can be argued that "feminism," while intended to combat the effects of sexism, basically promotes sexist ideas in reverse. I would never doubt or question some of the wonderful achievements brought about by the American Feminist Movement, or claim that these victories were anything less than good (such as the right to vote and the campaign for equal pay). But other than this, much of what we see from feminist mentality is an effort to divide men and women further, rather than bring them closer together. Feminism's intent is to win victories for women, but the main "achievement" of feminism is often to fight wars against men. It is now acceptable to ridicule, insult, and berate men, even to the point of denying them the very things that define their masculinity. Feminism rarely objects to this phenomenon, and many who consider themselves feminists heartily approve of this type of behavior as an expression of feminism itself.
God has created women for a very noble purpose: to help men. Being a man is tough. Women can openly show emotion and talk about their feelings with just about anybody. But society in general does not allow men this same luxury. The average man will tell you there his only one person with whom he feels safe enough to open up, and that's his woman. For this reason, God tells women to submit to men. It is not a command to be subservient, nor is this command equal with the term oppression. Women are commanded by God to respect and submit to men because this is how men feel loved. If you want to show a man you love him, you cannot love him the way a woman wants to be loved. You have to love him according to the way a man wants to be loved. Submission, therefore is an act of the will. You choose to submit. No man can force you to do it!
Women are commanded to submit to men. We hate this because we mistakenly think that this is a one-sided effort. But upon closer inspection, we see that men are commanded to love women. In my experience, men do a pretty good job of loving us. How about it, ladies? Do we need to be reminded of how they faithfully put up with our bad attitudes in spite of our ridiculous attempts at rationalizing our behavior? Simply say the letters "PMS" to a man and he will instantly back off. He will not defend himself against our rude, angry, unjustified cruelties toward him. He will just endure it. This is his way of loving us unconditionally, even when we don't deserve it. He does this every month for us, no? Yet we continually ask, "So you're telling me we should let the man do the job, even if he's a lousy breadwinner?" The answer is, "Only if you love him." If you don't love your man, go ahead and tell him how disappointed you are in him. Go ahead and take matters into your own hands, letting him know just how much of a failure you think he is. Feel free to respect him only when he deserves it. Just don't expect him to treat you with kindness the next time you are undeserving.
Scripture clearly commands both men and women to treat one another with kindness. Should women have "rights"? Of course! The Bible addresses this issue with men and commands them to treat us with love. Do men always treat us with love? Absolutely not. But this is no reason to launch a full-blown war against them. It is also no reason to punish every man for the crimes of one. That being said, it can be argued that there is no need for feminism at all. Why? The Bible makes such ideology unnecessary. A clear, thorough reading of the Bible will reveal that God does not hate women, nor does He consider them to be second class-citizens. He cares very much about how men treat us. (Consider 1 Peter 3:7, which states that if a man is unkind to his wife, God will ignore his prayers!) Instead, the Bible has made provision for the respect and proper treatment of women, declaring them equal in value to men (just different in role and function) as co-heirs of the Kingdom. No man-made (or woman-made) ideology is going to improve upon what the Creator has designed.