You don’t need to explain a mole. Ask them to make a sphere or cube out of 2-to-the-some-number of iron atoms. You can use a succession of pictures of atoms being doubled to communicate how many atoms to use.
You could use a unit of time based on the half-life of some specific isotope. E.g. how many half-lives of Uranium 238 does it take for light to travel the height of your body? Do radioactive decay rates depend on anything that could rationally be different on an alien world?
How many does it take for *your *body, pal ? And is that something you know off-hand ? 
It’s only tied to grammes for convenience. It’s really about a specific number of atoms or molecules of a substance. Assuming atoms are atoms on their planet, and they can count them (even if not very precisely), we’re good.
Read Arthur C. Clarke’s “Cosmic Casanova.”
I think this is a good approach. Instead of water on a surface where the surface properties might affect drop size, take a picture with drops dripping from a faucet, or moving through air. Even if the drops might vary in size depending on the source, you’d be able to estimate the true drop size from how ripply the drop is. I remember some old Sinbad movie with a boat going over rapids or caught in a whirlpool or something, and you could tell it was a model because the water didn’t look right.
If water’s surface tension properties vary too much with temperature or pressure, the triple point of water is well-defined. If you can, have them tell you pressure and temperature based on that. For one atmosphere, a pressure about 200 times that pressure. Pick a temperature a little above freezing, or some fraction of the way between freezing and boiling.
Ask the alien to take a photo of itself beside a diffraction grating experiment, ensuring the distance between slits is clearly visible.
(This works better if the alien is very small or if the resolution of the picture is very high).
Doesn’t the alien know how far away we are? Therefore we already have a common unit of length (albeit a very large one).
Get the alien to fold a sheet of paper seven times. I could be wrong here but I think that’s the limit and whatever size sheet you start with usually ends up at roughly the same size after the seventh fold.
prepares to be shot down in flames
Ka-pew! Nerrrrr…poom.
No, currently the second is defined in terms of that cesium oscillation, and the meter is defined in terms of the second and the speed of light. It amounts to about the same thing, though.
And standards like that are what I was referring to when I mentioned a scientist specifically trying to communicate size.
…use a banana for scale
Can we deduce anything from depth of field or camera settings? Or is that likely to be arbitrary also?
You could infer the field of view by looking at the local star: it’s going to be about half a degree in angular diameter. Why? Well, stars conducive to life only exist in a fairly narrow range: too large and they burn out too fast for life to evolve; too small and most of their output is in the infrared (which is not conducive to photosynthesis or the like). This also dictates how far away the star must be–the so-called Goldilocks zone where water is mostly liquid.
Unfortunately, this is as far as it gets you. You know the angular size of the alien but not how far away it is.
Here is a very indirect approach:
Creatures on Earth show a relationship between size and leg width. Small animals like bugs have thin, spindly legs. Elephants have trunk-like legs. This is because of scaling laws: mass goes up with the cube of length, but leg strength only goes up with the square. So big animals must have fat legs to compensate.
The two remaining variables are the natural strength of the creature’s chemistry and the local gravity. It’s reasonably safe to infer that the chemistry isn’t too far off from Earth life–we’re pretty well optimized as it is. It seems unlikely that Avatar-like life forms with carbon-fiber bones could really work.
Gravity is a harder problem. You might have a tiny but squat creature because the local gravity is very high. How to estimate this? One approach: piles of granular material form into cones that form a specific angle with the ground, called the angle of repose. This angle varies with gravity. If the picture contains sand dunes or the like, you could (crudely) estimate gravity.
I suspect that the real answer is that the aliens are going to be roughly our size (at least if we limit ourselves to beings that we even recognize as intelligent life). Cell-based brains probably can’t be tremendously smaller than our own. Larger brains could exist, but there are upper limits to a creatures size due to the aforementioned scaling laws. And we can’t just posit really low gravity, since gravity that’s too much lower can’t contain an atmosphere.
Of course I’m building in all kinds of assumptions. Maybe alien life is really weird. It might even be weird enough that size isn’t a meaningful concept.
I always think that when relatively humanoid aliens are posited. How do we know these aren’t 4 dimensional energy beings? And we’re only seeing a tiny aspect of them projected into a 3D world and then onto a 2D picture? Maybe they’re really a swarm of nano-aliens in a temporary cloud of hive like intelligence. Maybe they’re an eternal computation device made out of the random configurations of a set of rocks, and it takes them eons to think a single thought?
Aliens are always presented as just weird looking people. Although if they really want to freak us out, the writers make them weird looking insects or reptiles instead.
Ask them to stand next to a piece of lead that is thick enough to block 50% of the radiation that is emitted from Radium 226, or whatever gamma radiating isotope is the easiest to communicate.
Also ask them to point to their genetalia, lest we try to shake the wrong appendage when we meet them.
When I read the OP, I took it to mean that we couldn’t ask for more information, all we got was the picture. So no oscillating Cesium atoms, no diffraction gratings, no drops of water. What’s the best we could do in that case, following Dr. Strangelove’s approach or something along those lines?
I wonder, though–what kind of photo? We think of a photo as a rectangular grid of pixels, where each pixel contains the average flux across three separate frequency bands. These bands are set up to roughly cover the gamut of human color perception, which is only a narrow window of the whole EM spectrum.
But a really high-tech photo might contain a lot more information. An ideal camera would record the time, frequency, position, and angle of each incoming photon. We could do a lot more with this: it would act as a spectrograph, for instance, and we could use the resulting information to figure out the temperature of their sun, the composition of the local rocks, and so on. We wouldn’t need a baseline because atomic spectral lines are distinctive enough that we could figure out the units easily enough.
From there we’d have lots of avenues to explore. From the information about their star, we could compute the incident flux very easily, and from there reverse engineer their distance from the star. It might even match something in our planetary survey. We’d have a detailed chemical analysis of the surface, which would tell us something about the kind of material their planet formed from, and thus give more hints about the gravity. We’d be able to match their rocks with our own and look at the scale of wear patterns. If the camera was precise enough, we could use it as a microscope and look at the grain structure of the rocks and again match it against our own.
So all in all a fancy enough picture could tell us the scale of things with both high precision and confidence.
On the other hand, we might only get a single-channel low-res picture of some unknown frequency. We’d have a hard time inferring anything at all aside from crude scaling factors.
Well, there are a few writers that have pretty weird aliens. Reynolds’ Pattern Jugglers, Baxter’s Qax, and Egan’s planck-scale life are a few that come to mind.
Something I haven’t seen before: the holographic principle means that you can rewrite our physical laws such that we are really living on a 2 dimensional surface–however, we are irredeemably scrambled in that space. Something living on that surface would see us in 3-space as equally scrambled. Could the two ever contact each other?