The two examples Tyrion cites:
The Nights Watch
- They choose by popular vote on those who wants to be the next leader.
The Ironborne - (not 100% on this)
They choose via a Moot on those who want to be a leader.
Essentially Tyrion is saying you can't have children
but you could either pick a successor or let the people decide.
As to why he thinks it's important, he explains that: Your Grace, I saw hundreds of arrows fly towards you when you fought on Blackwater Rush, and
I saw hundred of arrows miss.
But any one of them could have found your heart.
Essentially if you keep flying into danger at some point danger will come your way.
Without a successor in place this whole thing will have been for nothing.
As @MikeScott mentions in a comment: He may also be trying to suggest to Dany that it's irresponsible of her to put herself in danger, since in fact there is no potential successor who can control her dragons (as far as anyone knows at this point), regardless of how they are chosen.
And without the dragons, the whole enterprise will immediately collapse.
Tyrion never outright states it
but it is certainly implied though I don't think it is his main reason for bringing it up.
As @Kepotx says in a comment: Also, if she win, but die without successor, that mean no queen, and new war for the throne.
This fact is not in her favour if she want to bring lot of people on her side.
If she is to gain more support having an available successor will help.
However, it doesn't seem to be common knowledge that Dany can't have children, and it's even debatable if she can or not, so this might not be the main motive behind Tyrion's comment.
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Tyrion is addressing both the immediate and future need for a successor.
In the short term, if she is going to expose herself by using and riding the dragons, there needs to be a clear successor to take up the reins
should she die, otherwise everything falls apart.
In the long term, it's important for the people to know that there is a stable line of succession, whether it be by birthright, or some other means.
Imagine that Dany does win the Iron Throne.
Great, right?
But what happens when Dany dies?
Presumably, she's not immortal.
When Dany shuffles off this mortal coil without a successor, we're right back to where we are now!
Various parties will be trying to scheme or force their way into power.
Look to real life history for examples of what happens when the line of succession is weak, or unclear.
Entire empires built by strong, charismatic individuals have been crippled or destroyed on their death because of this.
Look at the empires of Alexander, or Genghis.
Tyrion is advising Dany to avoid that fate.
At present, he's speaking to her about the short term ramifications, but you can bet he's thinking very hard about the long term as well.
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In story, it's because she's risking everything by placing herself in danger, as others have suggested, so I won't labour the point on that.
Dramatically, however, it's to foreshadow the events that followed, lending the later events a hint of cosmic irony .
Indeed, by doing as she did, 1 she appears to have greatly increased the risks everyone faces by losing a dragon to the night king -- and the situation could have been far worse ; Drogon might easily have been hit and she could have been killed -- in hindsight lending Tyrion's words some prophetic weight.
1.
(the spoiler is pretty major, don't say you weren't warned)
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He saw Daenerys fighting.
So he believed that she might die.
Once she dies, every thing will go in vain.
And that's why he didn't want her to go beyond the wall.
Besides, Daenerys can't be a mother, her dragons are her only sons.
So Tyrion thinks that she has to choose a successor.
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Even beyond the obvious facts that, herself being from a family of feudal rulers (re-)conquering a feudal realm, where succession is always massively important (which others have already touched upon), at this point it's also a simple military matter that would be just as important today.
You are fighting a war, it is absolutely paramount that there is always a quick and if possible simple answer to the question of "who is in command?
", if there isn't you can watch your forces crumble in disarray and confusion.
So far, this hasn't been a big problem, Dany has never been in a position of immediate danger during the battles that her armies have fought, but with her riding Drogon into battle, that has changed, so the question of "if something happens to you, who is in command?" becomes vital.
Especially considering that Tyrion has seen a lot of people in command die or get taken out during the last years, and he has seen the chaos that often follows, so it's only naturally for him to try and make sure things remain as stable as possible if something were to happen to Dany.
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In the episode Beyond the Wall, Dany and Tyrion had a lengthly conversation about successors. And Tyrion said to Dany:   You say you can't have children.  But there are other ways of choosing a successor.  The Night's Watch has one method, The Iron born folk, although many  flaws have another.   Why did he think it was so important to remind Dany of this now?What are the methods he is talking about exactly besides having children?
Why does Tyrion think a successor is so important at this point in Season 7, episode 6?
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The clearest reason that Tyrion thinks that a successor is important at this point in the series is because he believes that Daenerys might die. Daenerys can choose a successor as a way to avoid the chaos that will otherwise threaten if she dies childless
Daenerys can choose a successor as a way to avoid the chaos that will otherwise threaten if she dies childless