I'm a fairly new DM and we are playing through the Starter Set campaign (Lost Mine of Phandelver).

One of my players, who chose the pregen human fighter, wanted to punch a goblin with his bonus action after using his action to attack with his (two-handed) greataxe. Is this possible?

Not possible for the pregen human fighter(s)

In general, you only have a bonus action to use if a class feature grants it to you, or if you have a spell you can cast with a casting time of 1 bonus action.

The pregen (non-variant) human fighter is, well, a Fighter. At level 3, he gets the features of the Champion subclass.

The only class that can do a bonus-action unarmed strike by default at first level is the monk, who gets the Martial Arts feature at 1st level:

You gain the following benefits while you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield:

[...]

• 
  When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action. For example, if you take the Attack action and attack with a quarterstaff, you can also make an unarmed strike as a bonus action, assuming you haven’t already taken a bonus action this turn.

Fighters do not get this feature at 1st level. The only bonus-action attack they can do is through Two-Weapon Fighting, which any class can use:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

However, unarmed strikes are not weapons, and do not have any weapon properties (including the Light property). As such, there's no way for the pregen human fighter to make an unarmed strike as a bonus action.

Jeremy Crawford, rules designer for 5e, confirms/clarifies this here:

If I’m reading the PHB right, a player can NOT use “unarmed attack” as their second attack after a light weapon, because your unarmed self is not classified as a Light Weapon. Is that the correct ruling and reasoning? Thanks! :D

Your unarmed strikes have no weapon properties. As a result, unarmed strikes don't qualify for anything that requires a weapon property.

That seems illogical though that you can't stab someone and then punch them. It might be what the rules say, but I'd wonder if this might be a rules oversight.

If you’d like to attack with a weapon and punch someone on the same turn, you can do so easily with the Extra Attack feature. And that combo is even easier for you if you’re a monk. The two-weapon fighting rule has no bearing on this combo.

No

punch a goblin with his bonus action after using his action to attack with his dual wielding greataxe

I guess the player was referring to the "Two-Weapon Fighting" rules, PHB page 195.

The rules are:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.

A "light melee weapon" is a weapon with the "Light" property. A Greataxe has no "Light" property, and a punch is not a weapon at all.

If you allow treating unarmed attacks as "light weapon", some classes get a "free" attack, which would be abusing of the TWF rules. This is explicitly not RAI according to Mike Mearls:

Probably ok if players don't abuse it w/2-handed weapons