Sure, gladly. But it's not so much an argument as a simple observation, plus refuting or ruling out out a bunch of false notions of the good that people have gotten used to. And since it's an observation you see across all living organisms, there isn't any one piece of evidence or one specific argument. It's just part of the nature of life. So I think what is warranted is not an argument, but an example.
Think about a simple organism. Something without complex reasoning about higher values. Say, a frog.
What is good for the frog? What is bad for the frog? Well, the only thing that can matter to you, if you have the consciousness of a frog, is getting pleasure and avoiding pain. When you eat a fly, you are doing it for that reason. When you drink water, you are doing it for that reason. And so on. I mean, frogs probably can't reason at that level---but if they could, that is how they would have to reason about it. Pleasure is inherently valued because of the way organisms are, biologically. It is the thing that evolution uses to reward the behavior that is "wanted" (i.e. that "wins" in a natural selection sense).
In other words, natural selection only works if organisms have a reason to act. For extremely simple things (e.g. single-celled organisms), there is no "mind" or "consciousness" in the organism that regulates behavior in any way. But for complex organisms, there needs to be a mind. And the pleasure/pain mechanism is the way that natural selection has for regulating behavior in the minds of conscious organisms.
Pleasure and pain just are inherently valuable---they are the only things that are inherently valuable. Because of the way biology works.
It's no different for humans, really. Besides physical sensations, humans also have emotions of pleasure and pain. They have moral value systems. When your moral values are satisfied, you get pleasure; when they aren't, you get pain. That is not to say, any moral system is equally valid. It isn't. The best moral system is the one that leads to the most pleasure. That is the one that is in harmony with biology and reality, not religion, social conformity, environmentalism, or anything else, but proving this statement is too much for a comment, so I'm just putting it here so you get the gist of where I am headed.
To get back to my main point--one way to see that pleasure is the only inherent value is to try to think of some reason for acting other than that. Ultimately, there never could be one---not a valid one. But pleasure always can be a good reason for acting.
Keep in mind I mean pleasure in the broadest sense---humans are long-range creatures, so I mean joy, rapture, happiness, contentment, fulfillment, and so on, plus all kinds of physical sensations (good food, sex, etc.). So I am not advocating range-of-the-moment hedonism, like, coke and hookers. I mean, yes, if that's a way to maximize your pleasure---but it isn't for any human being I know of.