With the normal fighter builds found in the PHB, is there any weapon choice (hammers, light blade, heavy blade, axes, etc) that does statistically better damage based on weapon damage, power selection, and appropriate stat?
It depends on what you are looking for specifically. These weapon groups are generally the "best" for non-specialized builds. The other weapon groups do not have nearly the same power and/or feat support..
Heavy blades have the best proficiency bonus, this will lead to more hits and great DPR. As a fighter it will also help you enforce your mark by hitting more. Requires a moderate to high dexterity.
Axes - Requires a moderate investment in constitution. Not quite as much DPR as hammers but there are a lot of high crit weapons so your damage output will be spikier. (Doing 60+ on a crit can be a lot of fun)
Hammers - Requires a high investment in constitution. They will have the highest DPR in paragon levels and up thanks to Hammer Rhythm (does con modifier damage on miss)
Polearms - You will pay for the reach with a lower DPR per hit. However the increased reach may mean that you get more swings. The Paragon feat Polearm Gamble gives you threatening reach (more or less). Requires some wisdom.
Remember that strength and constitution feed the same defense (Fortitude). So if you use Axes or Hammers you are almost certainly hurting your Reflex and Will defenses compared to a fighter using a heavy blade or polearm.
Within each weapon group its pretty easy to spot the optimal weapon for yourself. Whether one handed or two handed, there is also a superior version in the Adventurer's Vault. The key is deciding whether you want to spend a feat obtain the superior version. Typically the superior version grants you +1 damage on average. If you are short on feats, its better to spend a feat on Weapon Focus (+1 damage/tier) than Weapon Proficiency. A striker should probably spend every feat possible to pump up their DPR while you, as a defender might wish to prioritize other things.
Let me rephrase the question so it focuses on equipment, rather than power choice:
Given the set of single-weapon level 1 at-will powers, which power will produce the best average damage per round (determined by a +3 proficiency weapon having 55% chance to hit, and a +2 having 50% according to the average defences per level mechanics)?
There are four components in answer to your question:
We can then consider these in light of synergies possible through feats and magic items.
Accuracy is the easiest component to deal with. Weapons have either a +2 or +3 proficiency.
The assessment to make is: "Does 5% of a weapons static damage outweigh equivalent damage at that level?"
When looking at damaging weapons, we'll only look at the superior archetypes (found in Adventurer's vault and dark sun):
Question 1: At which static mod does the fullblade contribute the same damage as the execution axe? (.55)(6.5+x) = (.50)(5.5+2+x) Wolfram alpha says 3.5. (That was... slightly counterintuitive) Therefore, at a normal strength mod of +4, a fullblade is superior to an execution axe, despite the latter's brutal 2.
Question 2: What are the item/feat combos (for a fighter) that result in superior damage over each tier?
Answer: It's possible to find something interesting in every combo. My personal preference in paragon is the Greatspear (does less damage but has reach) or glaive (counts as heavy blade which opens up Heavy blade opportunity) with polearm momentum and polearm gamble and forceful opportunist. Looking at the options in the Guide:
and so on.
There is a fighter build, by now, to suit any weapon choice, though most builds will be better with a decent idea of what you want to do in paragon (as the stat requirements can be a bit touchy.) I recommend choosing an at-will, one of the superior weapons, (1h or 2h) and then sitting down with the guide to figure out how to implement your choice. With a fighter, the only real requirement for damage is a superior weapon.
If you are talking about damage, Hammers will edge out other weapons due, at least in part, to hammers rhythm (as mentioned above). Keep in mind that a Str/Con fight will end up with two weak defenses which is often especially rough on a tank. Race can play a role in it too. If you are going to be a dwarf (a fine choice) the dwarven weapon training feat gives you a nice boost during heroic tier but that dwindles a bit with level. Also, are you going to be core forever, or is that what you're limited to at the moment due to funds, etc. Most of the weapon types that were weaker in PHB1 have since reserved some more feats to help with balance.
For heroic to paragon, core only, I'd go with a STR/con/wis dwarf with a shield and warhammer. That will give you the flexibility to swing it with both hands and get numbers close to a greataxe, but when you need the defense you can switch to hammer and shield. Dwarf Wep Training will keep you competitive at low levels and HR will kick in at paragon level. Dwarf plus a decent wis will give you an okay will defense and the shield will offset a less than amazing dex/int for your reflex def.
Statistically better for what? Some fighters are more focused on soaking hits, doing damage, repositioning enemies.. Weapon choice plays into all of that.
You seem to be asking about what will be the best damage-dealer though. Since you are asking about PHB builds only I'll leave off the Battle-rager for now (he gets a damage bonus when he has temp hit points) and the Essentials Slayer. If those were included, I think it's clear that the Slayer is the biggest damage dealer.
Here's one way to do it:
Great Weapon Fighter with max strength and the biggest two-hander you can manage (fullblade, and mordenkrad are good if you have the feat space to go exotic, greataxe is good for when you just want to hew.) Dragonborn is a nice choice. Weapon Focus and Fighter Weapon Specialization (it's a feat) actually stack, and another untyped bonus might be something like Savage Axe. Brutal Flnker is good too. Again, some of this isn't in the PHB.
Let's say Greataxe. Save your feats for buffing that up.
For Powers there are some that let you toss in Strength + Con bonus: Brash Strike, for example, and the one you really want is Wicked Strike. Neither of those are in the standard PHB, though. I think you are stuck with reaping strike and cleave.
There are other options as well!