That would depend on how handy you are.
My grandfather used to put aluminum metal sheeting on the floors of his outdoor hutches and then shavings over top. This won’t work well for you if you have harsh winters though and your bunny still insists on digging up the shavings; it would get cold. Aluminum doesn’t rust so you can wash it.
When I was growing up and my bunnies were outside, my dad cut plywood (I should say untreated before another member feels the need to point that out, but my guess is you already know that) and laid it over the existing wood floor of the run. That way if he did soil it, we could replace it for little cost. We usually kept at least a small layer of shavings over it as well though. The covered bedroom section of the hutch always had floor to ceiling shavings for the bunnies to make burrows in,
Another option is coroplast – the plastic signs are made out of. You can buy it at most graphic design/sign shops and it is usually rather inexpensive. Many people use it to build C&C (cube and coroplast) cages and there are a lot of amazing tutorials online showing how to shape it so that it has walls as well (so pee can’t seep downs between the wall and the wooden floor underneath). Just Google “C and C cage.” Again, it is washable and not too terrible if your bunny nibbles it. Plus it won’t get really cold in the winter like tile or lino.
Carpet depends on your rabbit. If he is a chewer, then I wouldn’t recommend it because he could ingest it. It will also soak up any rain or spilt water which could make him sick.
Since he is outdoors, I would highly recommend some kind of bedding on top of whatever flooring you choose. If you aren’t comfortable using shavings you can switch to Carefresh or another line of animal bedding. It could be the paper pulp kind or the shredded paper kind. This way if a chill creeps into the air, he can still stay warm. He may be digging the shavings away right now because its summer and he’s warm. But it is more expensive than shavings.