When the Era changes, new Great People will be of the new Era. (True for me on Deity; not sure the mechanism at lower levels of difficulty.). That means any available Great People of earlier Eras will be succeeded by a Great Person costing more.
If you play Single Player, you are player #1, so have the first opportunity to obtain any such last-chance-cheap Great People. If you have a lot of gold or faith, consider buying them before the AI does. And if you pay for more than half their cost, you get extra Era Score.
Note that the AI seems to buy Great People as soon as they can afford them. One exception I've seen is an AI not buying a Great Musician, I believe because they had no place to put a Great Work of Music. (Their capitol already had another Great Work, and it had no place to move that.). They might also delay if they feel certain to get it without buying, but I'm unsure of that.
The AI is not smart enough to delay buying a Great Person shortly before an Era changes, when they are the only one who could get it, in order to boost the Era Score of the next Era. The AI doesn't seem smart enough to avoid Era Score actions which won't help in time (or to rush some just in time to get a better Age).
If you play Single Player, you are player #1, so have the first opportunity to obtain any such last-chance-cheap Great People. If you have a lot of gold or faith, consider buying them before the AI does. And if you pay for more than half their cost, you get extra Era Score.
Note that the AI seems to buy Great People as soon as they can afford them. One exception I've seen is an AI not buying a Great Musician, I believe because they had no place to put a Great Work of Music. (Their capitol already had another Great Work, and it had no place to move that.). They might also delay if they feel certain to get it without buying, but I'm unsure of that.
The AI is not smart enough to delay buying a Great Person shortly before an Era changes, when they are the only one who could get it, in order to boost the Era Score of the next Era. The AI doesn't seem smart enough to avoid Era Score actions which won't help in time (or to rush some just in time to get a better Age).