Ouch, a new turn signal flasher relay is $65 at the local parts shop. Is it possible to DIY a repair of such a flasher relay?
Yes you can often repair turn signal flashers!
A variety of things can be wrong: here's one of them. Solder joints on the relay printed circuit boards are especially vulnerable to stress cracks, due to the clicking motion day in and day out.
Take the case off the relay and use a jeweler's loupe to inspect for hairline cracks, especially cracks those that go all the way around a pin that is shaken by the relay clicking. For example:
To repair get a soldering iron fairly hot, melt the existing solder until it is fully wet, release cleanly, and let the solder form a nice shiny fillet. This is a very easy soldering job even for beginners.