The earth could send out many cheap, robot probes to flyby nearby stars and evaluate them for further study. Advances in technology could be incorporated into new probes, before they were launched, so that the normal advance of science would not mean the generation ship project would have to be cancelled and redesigned every few decades. Each "discovery" probe could visit many stars on it's way out, (strung out in more or less straight lines and accessible by relatively fuel efficient maneuvers) until it was either lost or out of communications range.

If any stars were found that looked really promising, a more elaborate and expensive robot could be sent, with the capability to decellerate upon arrival and conduct a detailed exploration. Any really interesting worlds found during the exploration phase could then merit having a manned mission sent for a complete survey of the system.

This program would have the advantage that once it got rolling, reports from the first wave of robots would be arriving while new probes would be taking off. It would be an ongoing, long term, continuous program, with reports coming in and ships departing every year, for centuries; not just a one shot deal that takes centuries to get ready and then is never heard from again. Probes lost track of due to accident, social turmoil on earth (or alien action out in space!) would not cause the program to suddenly stop, because at any time there would be many of them deployed in all different directions, at different stages of their missions.

Assuming the launch of one "discovery" probe a year and subluminal speeds on the order of .1c, we would start getting reports back after a few decades, and over the course of just a century, we would have preliminary reports from dozens of stars and probably be launching our first wave of "explorer" ships. By the end of the second century, we'll probably have detailed enough information on a really interesting destination and we could launch our first manned "survey" ship.

Now that's a mission I might actually volunteer for.