Is my friend right about omitting $c^2$ in world famous tiny equation?
This is basically a philosophical question, but I'm going to take what will probably be an unpopular position that your friend's reason is basically wrong in the context of an introduction to special relativity.
Sure, you can work in units where c = 1, and then the equation $E = m c^2$ reduces to $E = m$. But that fact alone is kind of vacuous: you can also work in units $v = 1$, where $v$ equals 1 m/s, and then $E = m c^2$ reduces to the technically equally legitimate equation $E = (9 \times 10^{16}) m$. But this clearly seems like a "less right" thing to do.
In many contexts $c$ is the natural velocity scale to set to 1. But that because it's a highly physically privileged speed in special relativity, and in order to understand why, you need to understand a bunch of facts like $E = m c^2$. So (I would argue that) it's subtly misrepresenting the causation to say that $E = m$ "because" $E = m c^2$ and $c = 1$. I would instead say that $c = 1$ "because" $E = m c^2$ (and several other closely related facts).
The danger of setting $c = 1$ too early when first learning special relativity is that it hides the fact that $c$ does have a physical value, with a unique physical significance. It's not just a convenient simplification, like doing a mechanics problem where you assume that a car is traveling at unit speed. But once you're comfortable with special relativity at an intuitive level, then yes, you can absolutely say that $E = m$ and everyone will know what you mean.
Pedagogically, I agree with @tparker's answer in that I do think it is not a wise thing to rush to $c=1$ before a student is relativistically mature enough to not misunderstand it. However, ultimately, I think your friend is $100\%$ right, you can omit that $c$, and not just that, it's a bit silly to write that $c$ as an adult. ;)
A Wheelerean Delight
In Spacetime Physics, Taylor and Wheeler discuss a nice story. I will tell a version of it which is a bit improvised (read mutilated). Imagine a town where people didn't know how to build rulers. However, there were two rail lines in the town. One went North-South and the other went East-West. The NS rail line had markings on it each meter whereas the EW rail line had markings on it each foot. So, people invented two notions of distance: an NS distance which they measured in meters, and an EW distance which they measured in feet. However, one curious kid once figured out that if you took a stick, put it along the NS rail line, measured its NS distance, and then rotated it to align it to the EW rail line, its EW length would always turn out to be $3.28$ times its NS length. So, they had this nice formula $L_{EW}=fL_{NS}$ where $f$ was a universal constant of the town, measured to be $3.28\text{ feet}/\text{meter}$. Finally, an insightful kid came along and realized that lengths of sticks remain invariant under all rotations and thus the same stick can be used to define distances along any of the directions. So, he started measuring the NS distance and the EW distance using the same unit, meter. People cried, "Oh! the dimensions won't work out!", "This is just a trick!", and so on. But of course, each of those sentences is wrong. The kid had discovered that the heart of the concept of distances lies in that they are rotationally invariant and this allows us (in fact, forces us) to measure distances in the same units along all directions.
Coming Back to Question
While this is not an exact analogy, it's pretty close. In relativity, we learn that the speed of light is invariant, its value doesn't depend on the frame of reference used to measure it. This allows us to measure distances in the traditional units of time (and vice versa, i.e. it also allows us to measure times in the traditional units of length). Let's give an explicit example. Say, you know how to measure time. How do you utilize that to measure length? You can send a light signal along a direction and the time it takes for the light ray to travel a certain distance would be the value of that distance. Notice that it is incredibly important to notice that this is an unambiguous and useful way to define the unit of distance since the speed of light is invariant among all inertial frames. If you chose a sound signal to do a similar thing, you'd end up with an incredibly frame-dependent system of units where you'd always have to refer to some ground frame in which the speed of sound was supposed to be a certain value. In other words, it would not have been any actual simplification. However, with relativity, since it is ensured that the speed of light is invariant, we can measure distances in the units of time. What would this mean for, say, $\text{meters}?$ Well, since light travels $3\times10^8\text{ meters}$ in $1 \text{second}$, according to our new understanding, we can say $3\times 10^8 \text{ meters}=1 \text{second}$ because that's exactly the amount of time it takes for light to travel $3\times 10^8 \text{ meters}$ . Or, in other words, $c=1$ (notice that such a $c$ is dimensionless).
Some Hand-Waving and General Remarks...
While rotations mix all directions of space completely into each other, Lorentz transformations of relativity don't quite mix space and time into each other to the same extent. For example, you cannot Lorentz transform a timeline vector into a spacelike vector, etc. However, there is still sufficient unification of space and time so that there is no way to escape the notion of a spacetime continuum. For example, there is no separate invariant time interval between two events and neither is there a separately invariant spatial interval between two events. You can only have an invariant spacetime interval between two events. This also motivates the use of natural units, or geometrized units, where $c=1$.
Finally, it is incredibly important to notice that the physical importance of the value of $c$ is in that it is finite (rather than infinite). If the invariant speed is infinite, then our whole scheme of measuring space in units of time breaks down (as it would break down in Galilean mechanics because the invariant speed in Galilean mechanics is, of course, infinity). So, the fact that we can set $c=1$ is not a matter of a clever way of managing equations which one could have always done. Rather, it is the most succinct form of expressing the non-trivial physical fact that there is a finite invariant speed which allows for an invariant/unambiguous unification of the units of space and time.
Generally speaking, when there is a fundamental constant of nature which relates two quantities of different units, it's a sign that we should actually measure the two quantities in the same units, making the constant dimensionless. For example, in quantum mechanics, $[x,p]=i\hbar$ allows us to set up a system where $x$ is measured in $\text{GeV}^{-1}$, $p$ is measured in $\text{GeV}$ as usual (i.e., usual after setting $c=1$!), and this makes $\hbar=1$. If you don't use natural units, $\hbar$ would have had the dimensions of action (i.e., those of angular momentum).
Your friend was more wrong than right, as others here have said. I am just adding some thoughts that emerged after long experience with relativity. This becomes a question of practicalities. Basically there are pros and cons to natural units (units where $c=1$ among other things).
Pros
reduce clutter in formulae and in long derivations
train one's mind to see that some things are alike to one another (e.g. space and time) or identical to one another (mass and energy, if we mean rest mass and rest energy)
Cons
lose a nice way to keep track of physical dimensions (I have lost track of the number of times when $c$ came to the rescue and stopped me making a simple slip in a calculation)
eventually when you compare theory to experiment you do have to know either the value of $c$ in your units of choice, or else the length of your apparatus in units where $c=1$. Either way you cannot avoid finding out how the speed of light in vacuum compares with the lengths and times of the equipment you are using. It is simply wrong to think one can say "$c=1$" and leave it there.
The most convenient way to apply theoretical results to experimental observations in practice depends on the type of experiment. For Earth-scale observations of distances and time, SI units are convenient. For energies and momenta in high-energy physics, use mega-electron-volts and write things like $m = 0.511\,$MeV$/c^2$ for mass and $p = 2.1\,$MeV$/c$ for momentum. But note the way $c$ here appears in the units. (Eventually the expert will drop this $c$ but they know what they are doing; do not drop it if you are not an expert.) The most convenient way in astronomy is sometimes lightyears, but you'd be surprised how often astronomers use mega-parsecs.
The Earth is 499 light-seconds from the Sun. I always thought that was a nice friendly number, and remarkably accurate.