Some people have a custom of adding “Birshut Baal Habayit hazeh” in the Zimun. Can the wife be included?
1) The wording isn't an essential part of benching
הנוסח הזה הוא לא מעיקר הזימון
That wording is not the essence of the zimun.
The "birshus" is merely an addition. Sephardim, Ashkenazim, Yemenites, Yiddish-speaking people-- each of these people have their own versions for this portion.
2) מעיקר הדין, Women DO have an obligation of zimun
מעיקר הדין, נשים חייבות לזמן לעצמן כשיש שלוש נשים ופחות משלושה אנשים. וכשאנשים מזמנים, נשים עונות. אמנם נשים נהגו לא לזמן לעצמן משום צניעות (ע' שו"ע או"ח קצט, ו-ז. מ"ב שם. ערוך השולחן שם ס"ב. שו"ת תשובות והנהגות ד נא), אבל צניעות אינה מבטלת את עיקר הדין.
א"כ, לא נהגו להוסיף "ברשות אמי" או "ברשות בעלת הבית", אבל מותר, וכ"ש שבודאי אין למחות על אחרים שאומרים.
Women are obligated to summon themselves when there are three women and fewer than three people. And when men lead zimmun, women answer. It is true that women do not normally lead benching out of modesty (Shulchan Aruch, OC 199: 6-7, ibid. etc), but modesty does not nullify the essence of the halachah.
Therefore, while it was not customary to add "in the authority of my mother" or "in the possession of the landlady", it is permitted to do so, and all the moreso one should not protest against others who say it.
We had this topic discussed on a Shiur from R' Ben Tzur (IIRC) in our Kollel a couple of years ago. He said:
Because Zimun is a "stand-alone mitzvah", in addition to the obligation of Birkas Hamozoyn, it is a Zchus to perform it.
Because only one person can do that, it goes automatically to the host or to the most knowledgeable Rabbi (or a Cohen) according to rules of precedence. If someone else is Mezamen, he has to ask for an explicit Reshus (permission) from that person to have the merit of this Mitzvah.
Technically, if the Mezamen person is lower in precedence that several other people, all of them must be asked in that order.
Simce the host's wife does not have the right to be Mezamen, she is not/can not be asked for permission, just as the people lower than him (like his own kids) can not be asked.
However, eventually, people forgot the original reason for asking permission and imagined that it is a matter of appreciation (as @Danf said "Basically, it was a way of thanking everyone who was involved in preparing the meal") and started to add whatever they imagined. While it is not a Halachic problem on its own, it shows a lack of understanding of the reason for asking that permission in the first place.