Maybe Echo can confirm this, but as I understand it only the first is correct. So you can say 他有点儿忙, "he's a bit busy") but you would never say 他忙点儿.
That said, some adjectives are also verbs 开心. And you can using 点儿 after some of them when you're giving orders or talking about the degree of change. So you'll hear people say things like 开心点儿, where it means something like "cheer up". But I don't think you can use it with every verb, because there are some where it doesn't make sense.
I personally find the difference between verbs and adjectives so subtle in Chinese (which ones are just adjectives and which ones are also verbs) that for me a lot of usage boils down to just hearing what people say and trying to just internalize which you can use with which. If there's an easier way, hopefully Echo can share it. :)