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Playing the D&D starter set, I found a Goblin Helm after
the final boss in the cave dungeon (before Phandalin)
and I would like to know if it is even possible for it to give an armor bonus.
When my character found it, I was told it had a +1 AC modifier, but in a previous question (What is the function of AC?), I was told it may not be possible.
Is it possible?
The Goblin Helm as described is not an item in any of the core books, but that doesn't mean that it doesn't do exactly what your DM told you it does.
Generally, a helmet is considered part of the armour (eg, Plate Armour comes with a Plate Helm, which is factored into the AC), but again, the DM is always able to house-rule or create new items for the players to use, whether as a magical item that gives a +1 AC bonus as an enchantment, or a mundane item that is just sturdier and more protective than average equipment.
The only concerns a DM must take when creating new items, rules, spells, etc.. is game balance, and party balance. In this case, if it is non-magical, it is strictly better than an equivalent magical item with a +1 AC bonus, as it will never use Attunement, it will always function in an anti-magic field, and it can still be enchanted, on top of its mundane bonus. It would not benefit from the resistance to all damage types magical items enjoy however.
Additionally, giving out +1 AC early in the game can be considered a bit strong of a reward due to bounded accuracy, however this is not inherently a problem. (Some campaigns run high power with tons of magic items and rewards, see Monty Haul [TvTropes Warning!])
As long as encounters are balanced around this fact, everyone at the table is happy and having fun, and no one in the party feels underpowered as a result, you can continue to enjoy your sturdy goblin helm.
Sure, the DM is technically allowed to do anything he wants. He serves as the adjudicator of rules, and therefore is allowed to define new rules if he pleases.
Not without thoughtful consideration. Games are designed with a particular balance in mind. For low-level characters, like those you'd be using in the starter set, an additional 1 AC is a big deal.
While +1 AC only represents a 5% less chance to hit on the dice, but if a creature needs a 16+ to hit (which is possible with weaker creatures versus heavy armor), it reduces that creature's chance to 80% of it's normal ability. That can be significant.
There are also other considerations for this particular item. If it truly is non-magical, then it works in an anti-magic field and does not require attunement. That makes it better than some magic items.
Two examples are the Ring of Protection and the Robe of Protection
Ring, rare (requires attunement)
You gain a +1 bonus to AC and saving throws while wearing this ring. (SRD V5.1, p. 237)
The robe is described in the DMG, which I do not have handy at the moment.
There is another example, Bracers of Defense, but since they only work if you are not already wearing armor it may not fit your situation.
Wondrous item, rare (requires attunement)
While wearing these bracers, you gain a +2 bonus to AC if you are wearing no armor and using no shield. (SRD V 5.1, p. 212)
The helmet you have described with +1 AC is behaving the same way as the Ring of Protection: it is not normally a piece of armor, but due to magic, it adds to your armor class.
If the Goblin Helm isn't magic, the DM can still decide that it is a special item that adds to armor class in a way that other items do not. I did not find it in the Starter Set, so I'd recommend that you roll with it and accept the +1 to armor class.
As a DM I play the helmet giving +1 AC not stacking with Shield, so if you have a shield and a helmet you just get +2 from the shield.
