I kinda like finding strange or unfamiliar LEGO parts and trying to identify them on Bricklink... is that masochistic? Probably.
Occasionally I see a piece in a lot or posted online that I think is genuinely odd and proves difficult to find. Without relevant esoteric knowledge finding this part (for example) is a little bit of a challenge:

Unless you know it, or you've seen it in passing, have the set/s or happen to be a Bricklink Wizard, I think you'll be in for a bit of a search. Bricklink can be obtuse at times... and doesn't always show the parts explicitly (oo-err Missus!).
The part above is not in 'modified plates', and it doesn't come up if you search 'links' either.
Answer:
I only found
this by hitting upon the search term "2x2 plate" and being prepared to follow the results to the top of
page 4.
It got me thinking that here be the makings of a rolling challenge. To start it off, I nominate this part:

If you find it (or know what it is) please post a Bricklink part number, your search terms and a link to the part. Then it's your turn to set the next "Unusual Part" for identification and post a picture (no bigger than 600 pixels wide).
Some rules:
1. The part has to appear in the inventory of at least one set after 1958 (since the introduction of the 'system').
2. It must be LEGO.
3. Please post your answer seperately to your new nominated piece.
4. In the event of two people posting an answer at approximately the same time, the first post will take precedence.
5. Please avoid specific variants with 'a', 'b', 'c' suffixes. Although, I think minifigure and other printed parts should be okay.
6. Just a bit if fun, please don't take it too seriously, I'm thinking that this should be self-adjudicating. Comment and on/off-topic discussion is strongly encouraged.
PS. I won't be offended if you think this is stupid and don't want to play.
Comments
Seriously though, this sounds intriguing although likely to be well beyond my talent/time for searching. But I'll sure be interested in seeing the obscurities that might arise.
I know that one, it is part 969. A trailer stand.
https://www.bricklink.com/v2/catalog/catalogitem.page?P=969#T=S&C=9&O={%22color%22:9,%22iconly%22:0}
I knew what it was so just searched for trailer stand.
But that is maybe a bit too easy.
So here's my pic ...
I chose this as I found one about ten years ago when I was new to using bricklink and it took me about an hour to work out what it was.
Got it!
The part is decbase02.
I knew I'd never seen it before. So I searched via colour guide in Light Grey.
I selectively picked categories that I have had little cause to look in but that I know have more electrical or mechanical components in them. Train Track, brings up the part too.
I found it as you were talking about looking in electric categories in Light Grey!
Here's mine:
Colour brown, search for 2x6 and there are only 7 parts matching.
https://www.bricklink.com/v2/catalog/catalogitem.page?P=3795a#T=S&C=8&O={%22color%22:8,%22ss%22:%22UK%22,%22rpp%22:%22500%22,%22iconly%22:0}
I'd be more interested in how you find it. Remembering a set it is in is probably a quick way to find it. But if you didn't know that, how would you go about finding it.
https://www.bricklink.com/v2/catalog/catalogitem.page?P=x169&idColor=9#T=S&C=9&O={"color":9,"iconly":0}
I recognised it from an old Star Wars set which had Gungans in it so used the search term "Gungan" and eyeballed the set in the list.
I would have searched for light gray if I hadn't immediately recognised it.
I thought mine would be more difficult, given it was hidden in the Homemaker category!
Some parts that are very obscure can be quite easy to find if they have relatively simple functionality and you know the types of terms that would be used at BL. Whereas common parts that you use all the time can be difficult to find.
The joys of BL.
As to Dr Dave, it seems the lights are on but no-one is home! So let's put this one as his turn ...
I'd just popped on to the thread earlier to play and then had to go back to work, so thanks for filling in for me!
https://www.bricklink.com/v2/catalog/catalogitem.page?P=32952pb001#T=C&C=85
Found it by looking for “brick modified”...
How about this one?
I did think of this one piece which I had until recently but it’s probably very easy in comparison - it did only appear in 2 sets though:
and this...
Give me 10 mins or so to come up with a part...
I'll go for this funky looking thing.
Searched for 'Antenna' and it was halfway down the page :)
How about this one:
I thought that one was really good too! Searching with 2x2 won't find it, and neither will searching for Light Grey or Light Bluish Grey colours...
It's modulex, so the light grey colour has a mx prefix. As you rightly point out the non-standard dimensions exclude it from regular 2x2 results.
Good choice!
The part is Mx1642 Modulex Line Director.
https://www.bricklink.com/v2/catalog/catalogitem.page?P=Mx1642&name=Modulex Line Director&category=[Modulex]#T=S&C=125&O={"color":125,"iconly":0}
Your turn :)
So, here is another view from BrickOwl:
Simply searched for 'ball' :)
Try this one:
https://www.bricklink.com/v2/catalog/catalogitem.page?P=637#T=C
How about
Try this one:
Try this:
https://www.bricklink.com/v2/catalog/catalogitem.page?P=clikits111#T=S&C=12&O={"color":12,"iconly":0}
How about this one?
I'm off to continue watching the Red Sox baseball game. They're not so good tonight... :(
https://www.bricklink.com/v2/catalog/catalogitem.page?P=gal24#T=C&C=15
This?
https://www.bricklink.com/v2/catalog/catalogitem.page?P=x209pb11&name=Technic, Throwbot Visor with Silver Pattern (Sticker) - Set 8007&category=[Technic]#T=S&C=11&O={"color":11,"iconly":0}
I cheated and just searched "silver sticker"--it was on the first page
Ok, my last one before I need to head out...
Interesting use here:
by Joachim Klang