Following on from genius' thread plus a previous thread I can't find that asked for an option to block move reminders sent to them, consider a union of these principles:
Idea 1: reminder option appears when time remaining to claim win < threshold time
Idea 2: option for players to disable reminders so that they never get them.
Each player has an option in their settings "Automatically send me a reminder <time> before a game times out completely. A maximum of <number> reminders may be sent per day".
Or something like that. Limit <time> and <number> values sensibly. Either values as zero would turn reminders off.
I would expect the server load to be high so it would require some "intellegent" aspect to the polling procedure: e.g. Create a parameter "lowest time to timeout" per player. Create a list of all players with reminders turned on having "lowest time to timeout" > 3 days; another for 3-2 days; another for 2-1 days and a last list for less than one day. The "time to timeout" parameter is updated at a rate according to what list the player is on, ranging from daily down to hourly. Or something. Of course, if reminders were changed to internal messages only, or that option was available, then the server polling issue would be dramatically reduced, as it would only need to be tested on a player's games when they entered the site.