helst is the superlative form of gärna, which means gladly, and so this adverb goes like this, with respect to increasing degrees of "gladness": gärna, hellre, helst. They translate into English as: gladly, more gladly, most gladly; or similarly, preferably, more preferably, most preferably.
Saying vad som helst and substituting vad for x such that x is an object yields the following meaning: x such that it is most preferably. This translates into English as what is most preferable, which then becomes whatever one prefers, which becomes whatever.
Similarly, if you say vem som helst, you substitute vem for x such that x is a person. So, it yields the meaning: x such that it is most preferably = whom one prefers = whoever.
Here's an exercise for you to think about. Can you try and breakdown the meanings of the phrases var som helst and när som helst?
I hope that helps in explaining what it means.
In the sentence, "Jag vill göra vad som helst för honom," we use vad som helst and not någonting, because in Swedish, någonting may as well refer to something of quantity 1. If we had used någonting, the translation would be I will do one thing for him, which isn't any arbitrary set of things.
I hope that helps with the context. :))
It works, but there could be a difference in meaning. If you do something för honom, it'll be something he benefits from. If you do something åt honom it could be you do something so he doesn't have to, it could be you make something for him. It's a bit subtle but for the default English meaning of the sentence you'd want to say för in Swedish.
Moreover there's an expression göra något åt (stress on åt so it's an adverb/a particle) meaning 'do something about' which can cause misunderstandings here.