As a Fighter with an above-average Intelligence score, I've chosen to take the Magic Initiate feat (PHB, p. 168), and as my 1st-level wizard spell, I've chosen to learn burning hands, which I'm now able to cast once per day as if I were a Wizard. This fulfills the requirements specified by the Boon of Spell Mastery (DMG, p. 232), which reads:

Choose one 1st-level sorcerer, warlock, or wizard spell that you can cast. You can now cast that spell at its lowest level without expending a spell slot.

But there's a snag, in the fact that this fighter does not have spell slots to begin with—they are not an Eldritch Knight, and no part of the Magic Initiate feat specifies that they gain any spell slots. They weren't casting burning hands using a spell slot in the first place, so saying they "can now cast burning hands without a spell slot" doesn't mean anything.

Does this Boon benefit this character at all or is it useless to them?

The character would benefit from the boon

Magic Initiate says:

[C]hoose one 1st-level spell from the [class's] spell list. You learn that spell and can cast it at its lowest level. Once you cast it, you must finish a long rest before you can cast it again using this feat.

As you correctly say, Magic Initiate does not grant you spell slots, only (from this bullit) the ability to cast a single spell. So, you are already casting this spell without using up a spell slot. However, Magic Initiate only allows you to cast the spell once per long rest. Upon receiving the boon, you would be able to cast it as many times as you wanted. So there would be a benefit there.

Epic Boons - like the Boon of Spell Mastery - do not have any more complicated requirements than are indicated in each boon. Your Fighter's ability to cast Burning Hands is sufficient that they should then be able to qualify to take Boon of Spell Mastery upon exceeding 20th level by 3,000 XP.

All that said, Epic Boons are very directly to be given to a character at the DM's discretion and only with their approval, should the DM allow you to choose which Epic Boon you want. If the boon does not fit thematically (can you really be a Spell Master without having spell slots?) they are well within their rights to assign you a different boon, perhaps something like Combat Prowess more fitting for the character.