Websites think that both are acceptable.
However;
A friend of mine insists that “cleverer” is wrong, or at least bad, and that “more clever” is more correct 
Personally, I find “more clever” somewhat clumsy, because it takes an extra word to say the same thing.
This feels a bit like fewer vs less, but it’s not. Despite both being technically valid, I still wonder if one is still deemed better than the other, perhaps historically.
what do you think?
I’m pretty sure they’re both “correct.” Personally I’d tend to use “more clever” because I don’t like the mouth feel of “cleverer” but that’s a personal preference (hmm, and actually I don’t like either so I’d rephrase to use something else if at all possible) but Google Ngrams suggests that “cleverer” has always been more popular.
If something can be cleverest - which it clearly can be, “this is the cleverest post I’ve ever written”, e.g. - ipso facto it can be cleverer, too. It’s just that, as Josh says, when you try to say the word aloud the second half can dissolve into Rural-Juror style mush, so preferring “more clever” seems perfectly reasonable too.
That’s an excellent point - the spoken version.
"cleverer"is definitely a bit awkward to say. Perhaps this is the reason why some people prefer “more clever”.
When facing a choice, I always choose what sounds best, since that affects the reading experience for some.
Smarter. 
Do you mean correcter? 
Obviously using the French syntax with an extra word shows that you’re more clever than the peasants who use the native English syntax with a single word. That’s the whole point of using French syntax!
Historically, this use of “more” arose in imitation of French plus, and there was a tendency to use “more” with French-derived words and “-er” with English-derived words. That suggests it should be “cleverer”, since “clever” is an inherited English word. But this tendency has been well and truly confused over time, and now it’s mostly arbitrary which adjectives accept “-er” and which don’t.
(Latin once had a cognate to English “-er”: brevis “short” > brevior “shorter”. But in French this got eroded away and mostly disappeared.)
Wow, this is very interesting – thanks for posting it!
Ars longa, vita brevior doesn’t quite have the same snap, does it?
Fascinating, thanks.