Why would you not accept a free great person?

I have seen players on Twitch choose to not accept a great person they have attracted.

What might be a reason to do this?

Great Persons aren't actually free, but are purchased using Great Person Points (GPP). GPP act just like any other currency, and there's a different kind for every type of great person (i.e. great scientist points, great artist points, et cetera).

Once you've acquired enough GPP of a certain type you'll be prompted by the game to purchase a great person. If you buy it then all the GPP of the relevant type get deducted from your total (usually putting you back to 0), and the next great person of that type becomes available. If you pass, though, it's cost is reduced for every other civilization and you'll keep all your GPP. Once another Civilization buys the great person a new one becomes available for purchase, who you'll likely be able to buy with all those GPP you saved up from not buying the last one.

So, the reason you'd skip out on getting a great person would be because you don't actually want that great person, and would rather save up your GPP for the next one that shows up. The reason you might not want a specific great person is because not all great people are created equal, and some great people might be completely useless for you in a certain game (i.e. the great scientist which gives you bonus science for being next to natural wonders isn't any help at all when you first get it if you haven't discovered any natural wonders).

Seeing a lot of questionable answers...

Great people can be passed upon, it doesn't eat your points and you can either save for the next one (see the information below each GP) or you can spend money to get another one.

If you pass, you can still get said great person in a round or two.

I generally pass only when I see one coming soon that I really want. Aside that, I'll pile them up if I don't have space for their art or whatever until I build the correct district. But I looove tourism.