the first choice to turbo the ls is a common route that a lot of people take. the head of the ls motor flows significantly worse than the head of a gs-r or type r but the motor reacts well to a turbo setup. if you want to stick with this route, you may want to look into having the head ported and polished. the “standard” honda package from portflow is $1000 but will significantly help out the flow of the head. this may be your best bet.
swapping in an H series isn’t too bad of an option, but with the money that you’re going to spend doing the swap, you could already have a nice turbo setup.
swapping in a K series would be great, if they weren’t so damn expensive. you may be able to pick the motor up cheap, but the rest of the parts to complete the swap are going to kill your budget. if you don’t have a budget, this would definitely be a great choice.
another choice that wasn’t mentioned was to swap a gs-r head onto your ls bottom end. this is known as ls/vtec which i’m sure you’ve heard about. you can pick up a gs-r head for pretty cheap (under $500). there are a few changes to be made to the oil lines, but nothing major. this is a great choice if you want to stay NA. however, doing an ls/vtec setup also works well with a turbo setup. you could possibly do an ls/vtec motor and then turbo it for the same budget as a K swap. remember, that wouldn’t be the most expensive parts going into it, but you could still have a very reliable motor doing so.