There is an arguement in the gemara that says that if someone calls Avraham 'Avram,' they're either violating an asei, or a lo sasei (positive or negative commandment, respectively). How come this mitzva doesn't appear in the Rambam's count of the 613 mitzvot? Or anyone elses for that matter?

There are many more tzivuim in the Torah than 613 (if you were to make a count yourself you'd see), but we have a mesorah that there are taryag mitzvos. Those rishonim who made lists of mitzvos were trying to compile what they felt were the taryag mitzvos meant. If you think about it, the fact that one could count a mitzvah he found in the Torah that another didn't count demonstrates that there are more than 613 actual commands in the Torah.

True. BTW, where is the source that there are exactly 613 mitzvot in the torah? I know everyone says there is, but does the Torah itself say there are 613? What's the earliest source?

I'm sure Melech knows exactly, but whether the earliest source is an early midrash or a mishna or the gemara I am not sure.

It reminds me a little of the statement (from the Zohar?) that "yesh shishim ribuy osiyos la-torah", when in fact the amount is just half that. Of course taryag mitzvos is a bit different. Its definitely a given that there are "taryag mitzvos", however one counts them. That's not an esoteric concept the way 600,000 letters are.

Makkot 23b 4 lines from the bottom.
But R. Simlai, a third century CE Amora, isn't the earliest source. It's also in the Mechilta, Sifri, and various tannaic midrashim. It's an ancient tradition.
(If I recall correctly, the Pri Megadim in his introduction to his Shulchan Aruch commentary, the one recently annotated by R. Eisenberger, goes through some of the sources).

Your question still stands. Maybe the reason why no one counts this one among their taryag is because the machlokes about whether it is a positive or negative commandment is unresolved.

Not every opinion stated in the Talmud is considered as law!

There's a footnote in the gilyon hashas to look at Magen Avraham Siman 156. I don't have access to one, can someone look it up?
http://www.e-daf.com/daf.asp?ID=23 (bottom left)

The MA doesn't seem to clarify anything - he quotes the gemara but without further clarification.
(I couldn't find the other reference in the Gilyon HaShas, which I assume means the Pri Chadash at the end of siman 66 - I didn't see anything there relavent).

I don't have a gemara in front of me, but I believe that when I learned it I noticed a commentator in the back asked that. (maharsho?)

The maharsho says that it's mashma from both the Yerushalmi and Bereishis Rabbo that there is no machlokes, so that it would be an ase and a lo sasei. He also asks why it's not only not brought down as an asei or lo sasei, but isn't even brought down by poskim as an isur.

There is a machlokes haposkim whether calling Avraham "Avram" is an issur deoraisoh or just derabbonon (and the pasuk is asmachta be'alma). The poskim that hold it's mideoraiso also ask why it's not included in the list of taryag mitzvos and they bring down a Rambam from the peirush hamishnayos in Chulin (perek hey) that says that we don't count as mitzvos things which were given before matan torah.

Actually, it's the Maharatz Chayos, or however you pronounce it. He says something about mitzvos before Har Sinai.

Maharatz Chayes went to college. Bah.

*And* the maharsho. Get to the very end of the dibbur.