<PERSON>
The Kessel Run is obviously a major accomplishment that would make it known galaxy-wide. Just look to The Force Awakens. 30 years after The Return of the Jedi, <PERSON> meets <PERSON> and says:
This is the Millenium Falcon!? You're <PERSON>!?
And then later,
This is the ship that made the Kessel Run in 14 parsecs!
<PERSON> corrects this to 12, but it goes to show that this feat was widely known since Jakku and Tatooine are not close together. Jakku is in the Western reaches of the Inner Rim and Tatooine is in the Outer Rim (see: Map of Galaxy).
Side Note: some comments have mentioned that a smuggler probably doesn't want their ship to be known. This example shows the Millenium Falcon is known by name only. <PERSON> stole the ship from <PERSON> thinking it was a "garbage" ship and her only option for escape.
Legends Sources
The following was previously canon, but is no longer.
<PERSON> wins his choice of any ship on <PERSON>'s lot in Rebel Dawn and picks the Millennium Falcon by name. <PERSON> is described as being "dismayed" and <PERSON> is super excited to own it. Suggesting it has a reputation prior to <PERSON> owning the ship.
<PERSON> grinned, then threw both arms up into the air and whirled around in an impromptu dance, giddy with joy. Wait'll I tell <PERSON>! The Millennium Falcon is mine!
[...]
The Corellian felt physically tired, but so exhilarated that he knew he couldn't sleep - not yet. He had to savor his victory, his ownership of the Falcon, just a little bit longer.
<PERSON> makes modifications to the Falcon and begins taking other smuggling jobs.
Now that <PERSON> had the fast and (comparatively) reliable Millennium Falcon, he could take on the most challenging jobs. He still worked mostly for <PERSON>, who was basically running the Desilijic kajidic these days, but he also took jobs for other employers. The <PERSON> and his Wookiee sidekick became almost a legend on Nar Shaddaa as they broke speed records for the Kessel Run and flew rings around Imperial patrol vessels.
Note: this quote is indeed talking about speed records. 569
The book describes <PERSON> racing his girlfriend, <PERSON>, through the Kessel Run and beating her by fifteen minutes. It is only later in the novel that <PERSON> is desperate to escape Imperial vessels and goes "closer to the black hole clusters than any sane person would ever go."
Hey, <PERSON>. Look! He pointed at the instruments.
We set a record!
<PERSON> commented bitterly that their speed record had come at the expense of his nerves. <PERSON> eyes narrowed. Hey, this is weird, he said. It says we actually shortened the distance we traveled, not just the time. Less than twelve parsecs!
<PERSON> thinks the instrumentation must have shorted and was off. <PERSON> stubbornly believes he did it in under 12 parsecs. The novel ends with <PERSON> landing the Falcon in Docking Bay 94 to meet with <PERSON>, where it is found at the start of A New Hope. Based on the Rebel Dawn events, the Millennium Falcon seems to have a reputation as a fast ship. However, the second part of <PERSON>'s quote in A New Hope is likely him trying to make a name for himself and the ship, as he would have just recently made the record setting Kessel Run.
The proximity to the events of A New Hope do not match canon films like Solo, but <PERSON>'s boasting is in line with canon sources. In Star Wars Adventures #15, Flight of the Falcon Part 2, <PERSON>, <PERSON>, <PERSON>, and <PERSON> land on a planet in search of a new location for the Rebel base (since Yavin's location was compromised). <PERSON> is eager to brag about the Falcon doing the Kessel Run in 12 parsecs, not even its involvement in the Battle of Yavin.
It is possible through sheer determination on <PERSON>'s part that he has managed to makes his Kessel run infamous in the many years leading up to The Force Awakens.
True or Not?
It might be relevant to consider whether this feat was true or whether this is braggadocio, as comments have theorized. 605