A few comments...
First, in the same way as I'd not want to sell (or even give) a dodgy used car to a friend or relative, I'd also not give them an old router. Reason is, based upon personal experience as well as anecdotal evidence from the web, I believe the most common failure mode of routers from this century is (aluminium electrolytic) capacitor failure. The cap's used in pretty much all home routers were cheap-as-chips and often left the factory with an expected service life measured in months not years. As far as I understand, the failure process continues even when powered off and is accelerated by ambient temperature - I keep mine in the loft, where they get very hot indeed in summer.
As regards disposing of them, regardless whether given away or even recycled... if they have ever in the past been configured with your current WiFi credentials, and they fell into 'wrong hands', these credentials may still be recoverable regardless of any 'factory reset' or whatever. For that reason I tend to 'batch' mine for disposal and then, when I have a clear-out, I take the hit and change my WiFi password, reconfiguring all connected devices.
That said, negativity out of the way... I recently got a 'call for help' from an ageing Aunt, still on ADSL, with symptoms of a dodgy connection. My best guess, on what she has told me so far, is that her router's faulty. I therefor WILL probably dig out one of my old specimens, purely for the purposes of proving to her (and me) that a new router will fix things. Once proven, I'll tell her I want mine back then she'll have to buy a new router. But at least I'm glad I kept a couple back...
