The Insert Fasteners tool Fusion 360 groups fasteners into the Fasteners folder at the model's top tier, which offers very nice organizational benefits (as well as advantages in the Parts List in drawing environment). It would be useful if one could simply drag a custom fastener into the Fasteners folder, but I am echoing others in that suggestion.
Nevertheless, here is a work around to customizing that I have had success with. I am following @jhackney1972 advice from above. Here are my steps (with images) - I am converting a 3/8-16 nut into a 3/8-16 Locknut in my example.
1. Insert a similar fastener using Insert Fastener. I did not apply a joint at this step, since my geometry will change.
Note that breaking the link will remove the fastener from the Fasteners folder, so we do not want to break links as suggested above.


2. Locate the fastener in its stored directory and rename it. Note, it is probably best to move the fastener out of this directory altogether. Otherwise, the next time you insert the original fastener it will use the modified fastener instead.

3. Open the renamed fastener and modify it. In my case:
a. Delete all history from the original component start fresh.
b. Insert McMaster-Carr 90630A121 (step file without threads).
c. Drag Bodies to top tier component.
d. Remove (no delete) the now-empty McMaster-Carr component.
You can now save and update, but see Optional first.

4. [Optional] Clean-up component.
a. Turn off capture Design History
b. Turn on capture Design History. These two together clean up the timeline.
c. Add UNC 3/8-16 thread.
d. Save.

5. Update Design. I now have a locknut, which lives in the Fasteners directory. I can now joint as I see fit. Furthermore, I can duplicate the locknut using copy paste or Duplicate With Joints. These will also live in the Fasteners directory. The new locknuts behave nicely in the drawing environment, parts list.

This is worth mentioning again. If I leave the locknut (even though renamed) in its original directory and I later try to Insert Fastener with the original "Chamfered Bottom Hex Machine Screw Nut ASME B18.6.3 - 3/8-16 Steel Grade 2H Plain", what comes out is the Locknut. The fix for this is to move the locknut to the project directory.
Bill
Ripon Truck Repair & MFG
rtrmfg.com