tl;dr: Relativism bottoms-out in realism by objectifying relations between subjective notions. This should be communicated using concrete examples that show its practical importance. It implies in particular that morality should think about science, and science should think about morality.

Sam Harris attacks moral uber-relativism when he asserts that "Science can answer moral questions". Countering the counterargument that morality is too imprecise to be treated by science, he makes an excellent comparison: "healthy" is not a precisely defined concept, but no one is crazy enough to utter that medicine cannot answer questions of health.

What needs adding to his presentation (which is worth seeing, though I don't entirely agree with it) is what I consider the strongest concise argument in favor of science's moral relevance: that morality is relative simply means that the task of science is to examine absolute relations between morals. For example, suppose you uphold the following two moral claims:

"Teachers should be allowed to physically punish their students."
"Children should be raised not to commit violence against others."

First of all, note that questions of causality are significantly more accessible to science than people before 2000 thought was possible. Now suppose a cleverly designed, non-invasive causal analysis found that physically punishing children, frequently or infrequently, causes them to be more likely to commit criminal violence as adults. Would you find this discovery irrelevant to your adherence to these morals? Absolutely not. You would reflect and realize that you needed to prioritize them in some way. Most would prioritize the second one, but in any case, science will have made a valid impact.

So although either of the two morals is purely subjective on its own, how these morals interrelate is a question of objective fact. Though perhaps obvious, this idea has some seriously persuasive consequences and is not be taken lightly. Why?

First of all, you might change your morals in response to them not relating to each other in the way you expected. Ideas parse differently when they relate differently. "Teachers should be allowed to physically punish their students" might never feel the same to you after you find out it causes adult violence. Even if it originally felt like a terminal (fundamental) value, your prioritization of (2) might make (1) slowly fade out of your mind over time. In hindsight, you might just see it as an old, misinformed instrumental value that was never in fact terminal.

Second, as we increase the number of morals under consideration, the number of relations for science to consider grows rapidly, as (n²-n)/2: we have many more moral relations than morals themselves. Suddenly the old disjointed list of untouchable maxims called "morals" fades into the background, and we see a throbbing circulatory system of moral relations, objective questions and answers without which no person can competently reflect on her own morality. A highly prevalent moral like "human suffering is undesirable" looks like a major organ: important on its own to a lot of people, and lots of connections in and out for science to examine.

Treating relativistic vertigo

To my best recollection, I have never heard the phrase "it's all relative" used to an effect that didn't involve stopping people from thinking. When the topic of conversation — morality, belief, success, rationality, or what have you — is suddenly revealed or claimed to depend on a context, people find it disorienting, often to the point of feeling the entire discourse has been and will continue to be "meaningless" or "arbitrary". Once this happens, it can be very difficult to persuade them to keep thinking, let alone thinking productively…

To rebuke this sort of conceptual nihilism, it's natural to respond with analogies to other relative concepts that are clearly useful to think about:

"Position, momentum, and energy are only relatively defined as numbers, but we don't abandon scientific study of those, do we?"

While an important observation, this inevitably evokes the "But that's different" analogy-immune response. The real cure is in understanding explicitly what to do with relative notions:

If belief is subjective, let us examine objective relations between beliefs.
If morality is relative, let us examine absolute relations between morals.
If beauty is in the eye of the beholder, let us examine the eyes of the beholders.
…

To use one of these lines of argument effectively — and it can be very effective — one should follow up immediately with a specific example in the case you're talking about. Don't let the conversation drift in abstraction. If you're talking about morality, there is no shortage of objective moral relations that science can handle, so you can pick one at random to show how easy and common it is:

"Birth control should be discouraged."
"Teen pregnancy / the spread of STDs is undesirable."
Question: Does promoting the use of condoms increase or decrease teen pregnancy rates / the spread of STDs?
"Masturbation should be frowned upon."
"Married couples should do their best not to cheat on each other."
Question: Does masturbation increase or decrease adulterous impulses over time?
"Gay couples should not be allowed to adopt children."
"Children should not be raised in psychologically damaging environments."
Question: What are the psychological effects of being raised by gay parents?

I'm not advocating here any of these particular moral claims, nor any particular resolution between them, but simply that the answer to the given question — and many other relevant ones — puts you in a much better position to reflect on these issues. Your opinion after you know the answer is more valuable than before.

"But of course science can answer some moral questions... the point is that it can't answer all of them. It can't tell us ultimately what is good or evil."

No. That is not the point. The point is whether you want teachers to beat their students. Do you? Well, science can help you decide. And more importantly, once you do, it should help you in leading others to the same conclusion.

A lesson from history: What happens when you examine objective relations between subjective beliefs? You get probability theory… Bayesian updating… we know this story; it started around 200 years ago, and it ends well.

Now it's morality's turn.

Between the subjective and the subjective lies the objective.
Relative does not mean structureless.
It does not mean arbitrary.
It does not mean meaningless.
Let us not discard the compass along with the map.