How could a SYN flood affect a home router
There are three main ways a SYN flood can work against a home router:
If the router is performing NAT and has a port forwarded to a server, a SYN flood can fill up the router's NAT table, causing it to drop connections.
The SYN flood can act as a simple bandwidth-starvation attack. A typical home router is on an asymmetric connection with limited upstream bandwidth, so a SYN flood targeting a closed port can clog the upstream connection with RST packets sent by the router.
Home router firmware is often rather fragile. Simply throwing SYN packets at it too fast can cause a crash, taking down the connection.