Personally, I always assumed that we use m because it's the numerical multiplier on the independent variable in the slope-intercept form equation y = mx +b.
Michael Sullivan's College Algebra (8th Ed., Sec. 2.3), says this as part of Exercise #133:
The accepted symbol used to denote the slope of a line is the letter m. Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word monter. Write a brief essay on your findings.Of course, "monter" is a French verb which means "to climb" or "to go up".
But others disagree. Wolfram MathWorld says the following, along with citations of particular early usages (http://mathworld.wolfram.com/Slope.html):
J. Miller has undertaken a detailed study of the origin of the symbol m to denote slope. The consensus seems to be that it is not known why the letter m was chosen. One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is "monter." However, there is no evidence to make any such connection. In fact, Descartes, who was French, did not use m (Miller). Eves (1972) suggests "it just happened."The Math Forum at Drexel discusses this more, including a quote from J. Miller himself (http://mathforum.org/dr.math/faq/faq.terms.html):
It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In Mathematical Circles Revisited (1971) mathematics historian Howard W. Eves suggests "it just happened."The Grammarphobia site takes up the issue likewise, citing the above, and mostly knocking down the existing theories as lacking support. They end with this witticism by Howard W. Eves (who taught at my alma mater of U. Maine, although before my time):
When lecturing before an analytic geometry class during the early part of the course... one may say: 'We designate the slope of a line by m, because the word slope starts with the letter m; I know of no better reason.'To bring things full circle, I would point out that the English word "multiplication" is spelled identically in French (and nearly the same in Latin, Italian, Spanish, Portuguese, Danish, Norwegian, and Romanian), so lacking any other historical evidence, I don't see why m for "multiplication" isn't considered as a theory in the sources above. (Compare to k for "koefficient" in Swedish textbooks, per Wolfram.)