I will attempted to answer this question. Its a tricky question to answer because we simply don't have the information.

As far as I can tell, we have no references from early days of the mongol empire to earlier conquerers that we would know. The mongols had their own oral tradition and memories. They had their own heroes and culture.

However I can not say it is impossible for him to know some of these stories. The mongols met people who could have know about Attila as early as 1220 when they met people from Venice and quite possible others. However it seems unlikely that these traders knew about 700 year old roman writings (even if they were important for Christian History) and even less so that they shared this with the mongols (who were mostly not christian). The contact was not very extensive at that time.

So I think Atilla is a very unlikely candidate, Genghis Khan would probably never have heard of.

Another candidate would be Alexander the Great, as asked by u/spaceman_. Alexander had turned into a folk legend and their were many versions of him floating around all the medieval world, all the way from Europe to India. However its very unlike that the mongols would have known these tales during Genghis Khans lifetime. Once his sons and grandsons went deeper into Europe and the Middle East they surly would have learn of local legends and such, but they were probably not guiding them. They had their own justification to conquer the world, they did not need to imitate Alexander.

Their are a number of great Steppe Empires that ruled from the Eastern Steppe that are more likely to have left a lasting memory, but those would have been oral history for the most part.

The mongols early on had contact with the Uyghurs who had a form of written language. However I have never heard of historical works written in that script, or that the mongols read them. However Im certainly not a expert on Uyghur writings.

All in all the mongols had their own oral tradition, and I don't think we know much about what it was. Any conquerers known to us, were probably not known to the early mongols.

To expand upon the question, would the great Khan have known of the existence of Alexander the Great?

If not Atilla, would there have been some other historical figure that would be featured prominently in (hi)stories familiar to a mongolian or turkic steppe nomad chieftain?

There are theories that Atilla's Huns were originally from Northern China. French historian Joseph de Guignes argues for a hereditary link between the Huns and the Xiongnu. If that were true, it is much more likely that the Mongols would have known about the Xiongnu and possibly that a group of them moved West. However, it is unlikely that Genghis Khan knew about Atilla specifically. The Mongols simply had not reached Europe by then, they were en route but were called back when various Khans died and they needed to choose new ones. However, I would say it is totally possible that Kublai could have known about Atilla specifically due to ties to Italian merchant cities, whose messengers included Marco Polo. But, there is no evidence that the Mongols cared.