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.