(I initially posted this in February elsewhere, but have transferred it to here.)
If one were to open the Shulhan Arukh on the topic of saying the ברכת המזון (grace after meals) with a cup (for instance, see Berakhos 51a), one would find that Rabbi Yosef Karo wrote the following (OH 182.1):
יש שאומרים שברכת המזון טעונה כוס אפילו ביחיד, וצריך לחזור עליו, ולא יאכל אם אין לו כוס לברך עליו אם הוא מצפה ואפשר שיהיה לו, אפילו אם צריך לעבור זמן אכילה אחת; ולפי זה אם שנים אוכלים יחד צריך לקחת כל אחד כוס לברכת המזון; וי"א שאינה טעונה כוס אלא בשלשה; ויש אומרים שאינה טעונה כוס כלל, אפילו בשלשה
There are some that say that the grace after meals requires a cup, even for an individual, and he needs to go over it, and he shouldn’t eat if he has no cup over which to bless upon it if he is looking and it is possible that it should be for him, even if he needs to go past one eating time. And according to this, if there are two eating together, each one needs to take a cup for the grace after meals.
There are some who say that a cup is required only for three.
There are some who say that there is no need for a cup at all, even with three.
Rabbi Moshe Isserles added the following to it:
הגה: ומ"מ מצוה (ד) מן המובחר לברך על הכוס
In any event, it’s a preferable way of performing the commandment to bless over a cup.
There’s a lot there, but how’d it get to this?
There is nothing explicit in tannaitic texts, but there are two of them that later sages saw that there was something implicit in them – a mishnah in the tenth chapter of Pesahim (found in the Bavli on 117b) and a tosefta in the fifth chapter of Berakhos (this tosefta is found three times within the text of the Bavli – in Berakhos 52a, Pesahim 102b & 105b):
The mishnah
מזגו לו כוס שלישי מברך על מזונו ….
They mix for him a third cup – he blesses over his food…
The tosefta
הנכנס לביתו במוצאי שבת - מברך על היין ועל המאור ועל הבשמים ואחר כך אומר הבדלה, ואם אין לו אלא כוס אחד - מניחו לאחר המזון ומשלשלן כולן לאחריו
One who enters into his house upon the sabbath’s departure – he should bless over the wine, and over the light, and over the spices, and after that, say the separation prayer. And if he only has one cup, he should place it for after the meal and include all of them afterwards.
From the mishnah, Rav Hanan (in the Vilna printed edition) said to Rava that one can infer that the grace after meals requires a cup, whereupon Rava replied to him that the four cups the rabbis instituted out of the way of freedom (Pesahim 117b).
On Pesahim 105b, one of the redactors said that from the Tosefta, one requires a cup over the grace after meals.
From these two aforementioned Talmudic texts, the medieval commentators then develop the issue.
Maimonides codified (Hil. Ber. 7.16) that there was no need for a cup over the grace after meals.
However, Tosafos (Pes. 105b, s.v. Shema’ minah berakhah t’eunah cos) suggests that the halakhah is that it does require a cup (while mentioning that there is a custom of three blessing over a cup), even for an individual(!). (The Rosh (on Pes 105b) while offering a suggestion that goes against Tosafos’ reading of the individual language of the tosefta, he ends up bringing Rabbi Yehiel from Paris suggesting that a cup is necessary.)
The Ran suggested from the words of the Rif that it is a preferable way of performing the commandment, as it seems it is not necessary, similar to the Rambam.
(Hopefully this brief survey helps.)
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Tags: Grace after meals, cup of blessing, bentsching, birkat hamazon
Okay, so I keep trying to figure out what it is that I will say/type about my blog now that my blog has both been up for ten months now, has now received over 15,000 hits (thank you!), but more importantly, I get queried as to my blog - it's existence, it's point, my point, what I hope to accomplish (okay, so maybe that's a bit iterative), how I present myself, and other such questions. It gets kind of annoying sometimes being asked the same things over and over again, especially when I haven't organized a fully articulate or semi-articulate answer. As such, I realized that I should do something akin to a state of the blog post. However, I then realized over the past couple of days, that that would be a massive undertaking and probably better for both me to arrange my thoughts piecemeal in a topical fashion as well as for the readers (esp. the ADDers out there). (Don't worry, maybe at the end of July on my blog's anniversary, I'll do a State-of-the-Blog posting.) 
Unless one is speaking Ashkenazis (and "mezumenet" indicates otherwise), it should probably be "mezuman" (מְזֻמָּן) in the masculine, not "mezumen."
In Berakhot 43a and 46b, the מְזֻמָּן is the מְזֻמָּן לברכה, (written in the Talmud as מזומן) "the one designated (by the host) to say grace" (Jastrow, page 404) - the leader himself. Thus, מְזֻמֶּנֶת would be correct for the female leader.
If we'd like to justify the colloquial usage of "mezumen," I guess if we're in a situation where we're in a situation where we may/must perform זִמּוּן, then "we have [need for] a מְזֻמָּן," or "we [can] have a מְזֻמָּן," or "we have [already chosen] a מְזֻמָּן." Really, only once we choose the leader do we actually have a מְזֻמָּן.
Nevertheless, it seems to me that no matter who leads, the action is still זִמּוּן.
On the other hand, though, if one is going to mess up the meaning anyway, maybe it's better to use the pronunciation "mezumen" as an marker. :)
On a more important note, I've enjoyed reading your posts off and on for a while. You write interesting and throughtful!
I beg sliche for the pedantry. Perhaps Mar Gavriel has something to add.
-DJR