A brief search seems to indicate that no one is sure where the "Daddy" part of the name started.
Interesting (to me) geek-out info...
The answers to these questions depend, in part, on exactly which daddy longlegs you're talking about. That name is often used to describe several different creatures. For example, it may be used to describe the long-legged crane fly, which is an insect, or long-legged cellar spiders, which are true spiders.
Mostly, though, daddy longlegs is used to refer to Opiliones, which are an order of arachnids also known as harvestmen. Scientists believe there could be over 10,000 different species of harvestmen.
Harvestmen are arachnids, but not spiders. They have eight legs and look a bit like spiders, but there are several key differences. For example, harvestmen have one body section and two eyes, while most spiders have two body sections and eight eyes.
Harvestmen also don't produce silk and don't have venom. Thus, that old myth about daddy longlegs being the most venomous creature on Earth is just that: a myth. Even if you call a harvestman a daddy longlegs, there's no proof that its venom is especially toxic to humans.
Harvestmen tend to live on the ground in moist areas, such as under logs and rocks. Their long legs explain the “longlegs" part of their nickname, although no one knows for sure where the “daddy" part of the nickname came from. Curiously, scientists point out that there are some species of harvestmen that have short legs!
The long legs of most harvestmen are very important to them. Harvestmen sense the world around them with their legs, like we do with our ears, nose, tongue and skin. If you try to capture a harvestman, one or more of its legs might fall off. Although losing its legs might help a harvestman escape a predator, it seriously hurts its ability to sense the world around it.
daddy long-leg spiders
When people run into the creepy crawlies, the name daddy long-legs is used but this name can refer to at least three different animals.


Another critter often called daddy long-legs is a true spider referred to as cellar spiders in the family Pholcidae. This is a true spider with two body segments, and eight eyes, and makes webs with silk. This is most likely the animal you call daddy long-legs when you see them. Since they are most commonly found in cellars, hence their name.
There are also crane flies which is another creature that shares the well-used daddy long-legs name. These are insects in the family Tipulidae. They look like large mosquitoes but can't actually bite you because they lack a mouth. The stage you see crawling and flying around is their adult stage of life which only lasts a few days.
I will now crawl back into my nerd-cave and spend the evening doing more spider-WWWebing. 