Where does the story of the spider on the cross come from? I have a memory from childhood of being told not to kill spiders because there was a spider on the cross when Jesus was crucified. I cannot find any source in folklore (I doubt if it is anything else) and wonder if anyone else has heard this story before.

I am aware of the baby spider story related to the flight of Jesus -- a spider spun a web at the mouth of a cave to protect him and his family -- but I'm referring to a different story.
Okay, I did a lot of Googling using various search terms, all with &quot;spider&quot; and including one of &quot;Jesus&quot;, &quot;Christian&quot;, &quot;Christianity&quot;, &quot;Catholic&quot;, or &quot;Orthodox&quot; and one of &quot;myth&quot;, &quot;folklore&quot;, &quot;folktale&quot;, or &quot;legend&quot;. This is what I've found.
Web over the mouth of a cave
There are several versions of this one, but in every case, one party is fleeing from another and hides in a cave. A spider spins a web over the mouth of the cave, so the searching party assumes that their quarry is not inside because otherwise the web would have been broken.

David hiding from Saul (likely the original) - Wikipedia and spiderzrule.com
Mary, Joseph, and Jesus hiding during the flight from Herod - The Sutton Companion to the Folklore, Myths and Customs of Britain
Muhammed and Abu Bakr hiding from Quraysh soldiers - Wikipedia and spiderzrule.com
Saint Felix of Nola and Bishop Maximus hiding in an abandoned building. Twice! - Wikipedia
A World War II United States Marine - St. Luke the Evangelist Orthodox Church
Bonus: Yoritomo, a warrior from Japanese mythology hiding in a hollow tree - spiderzrule.com

Web over Jesus' cradle
This is where a spider spun a web over Jesus' cradle/manger and it protected Him from all harm.
I only found two sources with this: spiderzrule.com and The Encyclopaedia of Superstitions, Folklore, and the Occult Sciences of the World Volume 2 (twice).
The Christmas spider(s)
In this German/Ukrainian legend, spiders create cobwebs all over a Christmas tree and then something or someone changes the threads into gold and/or silver. This is believed to be the source of the traditions of hanging tinsel on the tree and putting a spider ornament on it.

Sunlight - Wikipedia
Father Christmas - spiderzrule.com (also mentioned on Wikipedia)
Child Jesus - kraftmstr.com (also mentioned on Wikipedia)

Spider on the cross
Unfortunately, I did not find any source with a story like this. In fact, the top Google results for &quot;spider on the cross crucifixion&quot; are almost all from this very website, Christianity Stack Exchange. If the tale does exist, I guess there isn't a record of it online. In that case, this question and answer is now the only record of it online!
So the answer to your question is (probably) nobody knows! If anyone does, they haven't said as much online (or in a book or article indexed by Google).