Big hole in front of mailbox from mail truck how can I fill it?
Not dirt, ideally. But you can use dirt if you can't get better material.

Better material would be crushed rock, ideally the sort that has all the fine dust from crushing still in it. Rock without fines can also work, but is less ideal.

Biggest problem with "filling a hole" most people have is the fact that you will lose (as you have seen) if you just put stuff in the hole you have. It will be saturated by any standing water, and it will be looser than the hole, and will get shoved back out of the hole easily with traffic.

So, let's look at what you could do differently for the poorest but most available fill - the dirt you have right there. You want to dig out the area of the hole you have to (at least) the full depth of the deepest part of the hole over the whole area of the hole, and somewhat beyond. Then you fill the hole with dirt - but not all at once. Put in an inch or two of dirt, and compact it throughly - don't use a large tamper (ie the common 10x10 square you probably don't own and would be wasting money to buy) - use something like a sledgehammer held head-down, handle vertical, and lifted and dropped over the whole area until it's all compacted. A 2x4 stud (inexpensive) is also effective used the same way (lift and drop on the end.) Then add another inch or two and repeat. In theory a high-heel dance party in high heels you don't mind getting dirty could also work, but it's not one I've tried. Physics says it should work, though. 

If you "upgrade" to crushed rock or gravel, you'll want to add a layer of (ideally, but difficult to get in small quantities) geotextile to keep the dirt below from "eating" your rock fill. Probably the closest thing to geotextile commonly available in small amounts is weed-block fabric or filter fabric. The idea is that water should drain through but your rock is not pushed down into the dirt below. Otherwise the tamping is similar. 

In both cases you do want to build up slightly, because tamping is good, but not perfect, and also because removing standing water greatly reduces the progression of holes from wheels, while allowing even a small puddle to develop quickly grows as water and dirt splash out. Running your own car smoothly back and forth over the patch (without stopping as the mail-truck does) will also help to compact it. If you do see it starting to puddle, fill it back in sooner rather than waiting for it to get worse (it won't get better by itself.)