I can imagine that somehow not responding to PINGs from WAN would be an ok idea regarding home network security. But can there be any down-sides to this?
I imagine games implement their own PINGs, so that shouldn't be affected. How about SSH connections and other, more super user-y things?
You should not have any issue at all by disabling "ping". The so called Ping is just another term for "IMCP Echo Request" or "ICMP Echo Reply" where ICMP is network protocol. If you "ping" something you send an Request to this machine. If the machine picks up such Request Package it answers with an ICMP Echo Reply. If you turn off "Ping from WAN" you simply tell your router that it should not care about any ICMP Echo Requests sent to you.
This also shouldn't effect any game. Hint: if the gameserver somewhare on the internet tries to ping you (for what ever reason the programmer might like to do this) the machine which is running your game won't respond because the router responds to this ping request.
Regarding your SSH question. It will not be affected in any way. SSH is another network protocol like ICMP is. This protocols aren't associated so that you can safely disable "Ping from WAN"
Depending on your router, disabling Ping can have a negative effect. If your router just ignores ICMP Echo Request packets, all will be OK as described by @Layticia.
On the other hand, if your router ignores ALL ICMP requests when you have disabled Ping, you could see several problems.
Your router must accept at least ICMP types 0, 3 (all codes), 4, 5 (all codes), 11 (all codes) and 12 (all codes). Dropping these types can cause your link to become congested, and can cause vital processes such as 'Path MTU Discovery' to fail. In some cases, if PMTU Discovery fails, you can see a very reduced throughput on your link, dropped packets and connections or other issues which would affect the performance and stability of your connection.
ICMP is vital for congestion control and status messaging between internet hosts. There is little security benefit to disabling responses to ICMP Echo Request packets, as there are other ways to determine whether a host is present on a give IP address.
See http://en.wikipedia.org/wiki/Internet_Control_Message_Protocol for information on ICMP. See http://en.wikipedia.org/wiki/PMTU for details of Path MTU Discovery.
