“I'll just let that one soak for a little while.”
You're doing the dishes. There's one baking dish that has food stuck to it and is hard to clean. You want to put it in the water and wait a while. You say this to yourself.
I'll just let that one soak for a little while.
The phrase "for a little while" means "for a short time". "For a little while" is more common and natural than "for a short time" in conversation.
Use "will" to offer to do something, or when you've just decided to do something like in these situations:
("I'll" is short for "I will", of course.)
When you've been planning to do something for a while, don't use "will". Say "I'm ___" or "I'm going to ___".
When you "let" something happen, you simply wait for it and don't do anything. Here are some verbs that English speaker use with "let":
let it cool
let it dry
let it simmer
"Soaking" something means putting it in water and waiting for the water to come into it.
You soak dishes in soapy water so that the food that's stuck on the dishes starts to come off.
You can also soak (in a bathtub) to relax and soak dried beans before cooking them.