Solo: A Star Wars Story introduces us to Han when he's around the age of 17-years-old, a gutter rat working for Lady Proxima, desperately trying to escape Corellia with his girlfriend Qi'ra. But how did Han wind up like this? At first, the movie doesn't seem interested in telling fans the story of baby Han before he was orphaned. But a heart-to-heart with Lando over the Millennium Falcon reveals Han remembers his parents, especially his father. Who is Han Solo’s dad? Warning: Spoilers for Solo: A Star Wars Story follow.
First a little history on the Falcon. The ship is a YT-1300 Corellian freighter, originally created by engineers from Han's home planet. The ship's age is lost in the mists of time. Wookieepedia lists it as being commissioned "more than 90 years before the Cold War," which began about a year after the events of Return of the Jedi, in 5 ABY*.
*(ABY = After (the) Battle (of) Yavin, aka the Battle of the Death Star at the end of A New Hope. The timeline of events which happen before (the) Battle (of) Yavin are BBY.)
So the Falcon was commissioned around 85 BBY, decades before the events of Star Wars: The Phantom Menace. In short, it is a very old ship. It was running for years before Lando got a hold of it, the kind of freighter Corellia has been putting out for generations before Han came along.
Han may not exactly be the kind of guy who spends his life on one planet, but he does have enough of a soft spot for his Corellian heritage. He instantly feels the Falcon was destined to be his. More importantly, it is a ship that ties back to one of the few memories he has of his father.
In the original Extended Universe novels, which Disney has now moved to "Legends," Han's father was Jonash Solo, son of Den Solo, and the scion of a now-defunct line of Corellian royalty, crushed under the Imperial invasion. He and his wife Jaina died when Han was young. By the age of seven, Han had been sold by his uncle to Garris Shrike, a smuggler who used young children to panhandle, steal, and run illegal jobs for him, not unlike Lady Proxima in the film.
The film writes out the House of Solo canon completely from the Star Wars universe. Not only do they take out the royal lineage Han is descended from but his last name is now just a bad joke bestowed on him by an Imperial Recruiter.
In the new canon, Han's father was a wannabe pilot, just like his son. But unlike Han, who escaped Corellia and followed his dreams, Han's father did right by his wife and child and, like many have done before him, got himself a dream-adjacent job. If he couldn't fly the damn things, he could build them. Those are the memories Han has of his father from his childhood days:
My dad worked the line at the CEC plant before he got laid off...He wanted to be a pilot.
The CEC plant made YT-1300 Corellian freighters, just like the Falcon. During the years his father was gainfully employed, ships just like this one are what Han remembers his father working on.
Getting the Falcon for Han wasn't just important because he felt like flying a Corellian freighter was true to his background and upbringing. It was getting to live out the dream his father never could, a touching homage to the family he lost at such a young age. Turns out Qi'ra right about him being a good guy.