So long as we're not bumping into a logistic growth ceiling, I don't see a strong incentive to leave our solar system
Along with generation ships comes the assumption of a non solar power source and the technology to engineer closed ecosystems-- the Main Asteroid Belt would be open to real estate development.
I see several logistic growth ceilings here. I suspect it'd take centuries or millennia to fill the Main Belt. From there the Hildas are natural cyclers between the Main Belt and Sun-Jupiter Trojans. (See this vid where first half shows <PERSON>, 2nd half Trojans). When the Main Belt and Trojans are filled up, there's the Centaurs as well as moons of gas giants. After those are filled comes the Kuiper Belt. And then the Oort Cloud.
By the time we fill the Oort, our civilization will probably have many millennia of experience building biomes from ice balls as well as using fusion power.
Given a civilization with this infra structure and capabilities, I'd construct the generation ship in the Oort. On the boundaries of the the sun's gravity well, it only takes a small burn to drop the ship to a perihelion deep in the sun's gravity well.
It'd take a few centuries for the ship to fall from the Oort to the inner solar system. 99
But given city states throughout the Oort, spending centuries dwelling in artificial biomes is already the norm.
When 1 A.U. from the sun, the ship would be traveling about 42 km/s. If perihelion is within .1 A.U. from the sun, perihelion speed is about 133 km/s. Doing the burn near the perihelion can give a substantial Oberth benefit. If the ship does a 55 km/s burn near perihelion, it'd pick up 133 km/s Vinf.
Of course 133 km/s is only .0004 C (speed of light). It would take around ten millennia to reach Alpha Centauri and longer for other neighboring stars.
If the ship had a drive capable of reaching .1 C, solar Oberth benefit would be of little consequence. In this case I might depart directly from from an Oort object a light year or two out -- as close as possible to the destination star. 99