Fame and fortune are what most adventurers set out to achieve. Very few, however, actually earn it. But the point of playing D&D is because it's fun. Since fame and fortune is perceived as "fun", it makes sense to letting the players get known for their deeds and maybe earn some gold along the way. In fact, Wizards of the Coast themselves openly support fame(as shown ever since they introduced renown points)!
You can't just give out renown points whenever you feel like it, though. There are many consequences if the players have a large amount of renown points. Sure, abuse is one matter, but we covered that in Abuse of Power(which should be a few posts before this one). What is the real matter is, well, the most obvious one.
I assume all of us know the famous Genghis Khan(if you don't, pay more attention to your history lessons). When we ask people, "Do you know who is Genghis Khan?" they know. Similarly, the PCs should be just as famous as even though they may not have led a rampaging army across China, they probably did save a few(a few hundred thousand, that is)towns from the evil Warlord/Tyrant/Dragon/Beholder/Zombie King/Other super evil and powerful monster etc etc, and will probably be just as famous as Genghis(imagine this: one day you walk into history class, and your teacher Mrs Smith explains that today we will study Bob the Dwarven Fighter...).
The PCs should, in my opinion, also be awarded renown points if they show off their trophies(as mentioned in the post just before this one, "Trophies and Rewards"). Once you let them know that they will be actively rewarded fame, they will be much more likely to try and score renown points(and then show them off in the real world and gain even more renown points there, which will lead people to inviting them to join their gaming group-I call it the Snowball of Fame effect)