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> 彼のこと **で** 何かご存知ですか  
>  Do you know anything about him?

I'm not sure what で is doing in this sentence. It's mentioned in [this
thread](https://japanese.stackexchange.com/questions/80414/double-checking-my-
understanding-of-%E3%81%AE%E3%81%93%E3%81%A8%E3%81%A7) but not broken down in
detail.

I'm familiar with Xのこと as "about X" but have no idea about the で part. Maybe
it has something to do with the keigo? Obviously we need a particle here, and
I'd like to use を, but cannot because there is no verb.

If I convert to less formal speech, is there any difference between the
following:

> 彼のことを知っていますか  
>  彼のことで知っていますか  
>  彼について知っていますか
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question
grammar particle-で keigo
Use of で in 彼のことで何かご存知ですか
No, it doesn't have to do with your example sentence being in 敬語 and the verb
in 尊敬語. Note the [link](https://www.edewakaru.com/archives/15531056.html)
@broccoli facemask provides in their
[comment](https://japanese.stackexchange.com/questions/80414/double-checking-
my-understanding-of-%E3%81%AE%E3%81%93%E3%81%A8%E3%81%A7#comment131150_80414).

> 【接続】  
>  名詞［辞書形］＋のことで
>
> 【意味】  
>  **〜について**
>
> 【例文】  
>  ①N１の文法のことで質問があります  
>  ②留学のことで相談したいんだけどいい？  
>  ③A：春休みに行きたいところはある？  
>  B：そのことで話があるんだけど…  
>  ④お金のことで話さなきゃいけないことがあるの…

You don't see を in there for two reasons:

  1. 彼のことで ＝ 彼のことについて
  2. を is omitted and if put back should mark 何か

> 彼のことで/について何か(を)ご存知ですか

何か is the content (内容) of the action "to know", and 彼のこと is the scope (範囲).
Literally: "What do you know about him?" Here "what" is the direct object of
"know", not "about him". See? English works the same way and also allows us to
set a range and talk about the specific content with one verb plus some
prepositions.

Let's take a quick look at your sentences at the end:

> 彼のことを知っていますか

This asks directly and specifically if the listener knows "him" ( _or_
something about "him" that the asker has in mind). Depending on the context it
could either be "Do you know him?" "Do you know about him?" or even "Do you
know what happened to him?"

> 彼のことで知っていますか  
>  彼について知っていますか

These two sentences mean pretty much the same. Standalone and with nothing
else, they sound a little strange to me.

> 何か彼について知っていますか  
>  何か彼のことで知っていますか

sound more natural to me.
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