What is the proper response when someone tells you, for example, that their mother has died? Relatedly, what should you do to express your sympathy? Do you send a card? flowers? bake a casserole?
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There is a common phrase for that.
Regarding sending something, there is a special custom in Japan called こうでん「香典」 - giving money to remaining family members with the purpose of offering it to the departed soul. (Originally, this was used as an alternative of こうか「香華」 - incense and flowers)
In my experience, I have seen this two times: one for a colleague's mother, and one for someone from another department. In both cases, another colleague started to collect money and put it in an envelope to give to them.
And there is another custom, こうでんかえし「香典返し」, which refers to getting back some present/food from them.
"I'm so sorry to hear (of your mother's death)"
Other users have already provided good answers, but here is mine anyway. When attending to a funeral, the most common thing to say is: ご愁傷さまです (go shuushou sama desu)
If you are writing an email, you can also say go shuushou sama desu, but as @istrasci pointed out you can also say things like 心よりお悔やみ申し上げます (kokoro yori okuyami moushiagemasu), though I wouldn't know if is ok to say at a funeral since it is sounds very much like written language.
Regarding what to send... I'm sorry I have to research that further.