Technically both phrases are correct. It mainly depends on how your going to say it. It's more commonly used as " half an hour"

Both are correct but the second one seems more smooth :)

"He's half an hour late." is the most correct, in that it is used in most of the English speaking world. However, "He's a half hour late." is used in parts of North America.

I think they are both OK. Most people I know would say "He's half an hour late." (I am in the south of the UK). I think "half an hour" is more useful generally - e.g. I think it would sound strange to say "let's spend a half hour tidying up". But other people might disagree!