רַבִּי יְהוּדָה הָיָה נוֹתֵן בָּהֶם סִמָּנִים: דְּצַ"ךְ עַדַ"שׁ בְּאַחַ"ב. We call the last Makah "Makas Bechoros" - why didn't Rabbi Yehuda say בְּאַחַ"ם
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This Haggadah (citing Yesod Hatorah) says that in fact R' Yehudah, with his acronym, is implicitly disagreeing with the other Sages.
A Midrash (cited in Tosafos to Shabbos 87b, ד"ה ואותו) tells how when the Egyptian firstborn heard from the Jews (on Shabbos Hagadol) that they were marked for death, they rioted to try to force Pharaoh to free the Jews so that they would be spared, and many Egyptians were killed. The name מכת בכורות hints at this: it means "the smiting [accomplished by] the firstborn." (According to this approach, I suppose, the deaths of the firstborn five days later, on Pesach night, is subsumed under this.)
Whereas R' Yehudah disagrees. He says that the firstborn did not wage war against the Egyptian government, and that the tenth plague was just that - Hashem killing all of the firstborn. So he uses ב (standing for בכורים) for the last makkah, rather than מ.
Aside from this, there are a number of reasons given for R' Yehudah's acronym. A simple one is just that he was in the habit of doing so in other places too (for example, Menachos 11:4).
All the Makkoth could be called "Makkath _". It could just as easily be called "Bechoroth" as "Makkath Bechoroth". The ב fits as well as all the others. The fact that it is referred to in the Haggadah may just be a reflection of the common way it is referred to in the literature and by the general public. That, in turn, could be a reflection of people's (and I include HaZa"L in that) reluctance to just refer to it as "Death of the First Borns" or merely "First Borns" without first appending the title "Plague of __" beforehand to distinguish that they are referring to a very specific incident that only occurred at one time and only in Egypt to the Egyptians at that time as a matter of punishment/Middah KeNegged Middah.
The fact that the Siman leaves out the Mem is probably because he's trying to shorten it; if he shortened it to Mem-Beith it would defeat the purpose. Why not just Mem (without the Beith, as suggested in the original question) could be as I mentioned above, that it's too generic; or see my question below - I wonder if the newly formed words from the Simanim meant something.
The real question is why the Simanim? What do they mean, and why are they more memorable than just the full names of the plagues (assuming the point was to make it easier to remember them)?