As a person who has never walked by a penny smasher without smashing a penny: Any old copper penny is fine. Pick the cleanest one you have.
Maximum clarity is achieved with pre-1982 pennies. It can be well-worn or have nicely-defined features; that doesn't matter, those features are getting smashed anyway. (Post-1982 pennies are primarily zinc and tend to produce smeary results that have their own way of being interesting, but that's not what you asked about.)
Sometimes, the grime on an old coin gets in the way. You can wash it first. Take it in the bathroom that's always nearby at these touristy places and use soap and water; get it spruced up a bit.
You can even polish it. Coin collectors hate the idea of chemical polishing but they aren't invited to this coin-smashing party anyway. There's lots of different things that can be used to make copper shiny and bright and some are very easy to find while out and about.
Like ketchup, from the burger joint right over there: Just smoosh some ketchup around on a penny that's already fairly clean and the acids in the ketchup will do a decent job of making it bright and pink over the course of just a few minutes. (This works on copper-bottomed cookware, too.)
If it's one of those new-fangled machines that takes credit cards and comes pre-loaded with pennies, then that's fine: Smash some pennies and clean/polish them later.
I'd imagine that pennies minted before 1982 (those that are mostly copper) are best for these machines. The newer pennies (now discontinued R.I.P.) probably aren't great under pressure.
The new machines I’ve used lately, my kid is into it, take credit card and have the penny preloaded
It’s probably more expensive usually $3-5 but also, it’s usually tourist trap pricing. I never have carried a penny on me since probably the 80s when I could actually use it to buy gum
Best to use an old solid copper one, for the USA that's before 1982, for the UK before 1992.