No, that wasn't exactly how I meant. Sorry If I'm being unclear.
Yes probably på puts the emphasis in the sentence on the boy's autonomy (på -sig själv-)
"Klä" is related to the word "kläder" which means "clothes".
"på" means "on".
So "klä på" essentially means "put clothes on".
The word "sig" specifies whom the clothes are put on.
The word "själv" is referring back to the boy, and means that he's doing it "by himself".
"Pojken kan klä på sig själv."
"The boy can put clothes on himself, by himself"
"Pojken kan klä på sin syster själv."
"The boy can put clothes on his sister by himself"
while in leaving the particle out we have this emphasis on the act of dressing/putting clothes on.
I think when we leave out "på" we putt
less emphasis on how the clothes are put on.
When we talk about how somone is dressed we normally leave out "på".
"Han klär sig så fint" normally means I think he has nice clothes on.
"Han klär på sig så fint" might sound like it is the act of putting clothes on that I think looks nice.
If I think a sweater looks nice on you I can say that it "klär dig".
"Tröjan klär dig" - The sweater looks nice on you.
"Tröjan klär på dig" - I guess this could work in some situations, if you get the stress right, otherwise it might sound like the sweater is helping you getting dressed which is wierd.
When we "dress" an object, by decorating it or putting something over it, we seldom use "på" (sometimes we use other prepositions).
"Klä julgranen"
"Decorate the Christmas tree"
"Klä på julgranen" is not idiomatic.
One last question though: What would you rather say more often, do you think?
It depends on what I was trying to say. In your example I would use the second sentence ("Pojken kan redan klä på sig själv.") assuming I mean he is able to put his clothes on by himself. I have a hard time imagining a situation where I would use the first sentence ("Pojken kan redan klä sig själv") but I might say something similar like "Han vet hur man klär sig" if he shows an ability to choose appropritate clothes for the occation (weather, party, or whatever the cirumstances are) but this says nothing about his ability to but them on by himself.