So, its important for a review of a card to address the merits and detriments of a card, so lets get that out of the way. As an economy card, we have to compare it against the OG Emergency Cache. In order to get as many resources for the same number of actions, it has to absorb at least 3 points of overkill - that is a LOT of 'wasted' damage. It also takes time to reach that level of overkill, unless you're going all in with Shotgun or hitting a Swarm of low health enemies with Mk 1 Grenades, Dynamite Blast, Storm of Spirits, etc. (and pushing forward all the excess damage to super-overkill the host). If you draw it late in a game, you might not have enough overkill opportunities left, and unlike the Cache, you can't find this early with Stick to the Plan or Backpack.
Much of the time this can be slow, awkward, and potentially weaker ECache. It has a couple skill pips for a flexible commit in a pinch, but otherwise its worst case scenario value is much worse, but its best case is, at least in theory much better. Letting a roguish fighter with a boomstick (Sawed-Off or otherwise) Double or Nothing to blow an enemy to smithereens and walk away with a juicy payout. However, in this case, you're probably just better off using "Watch this!" and getting enough.
However, there are three Guardians who deserve special recognition.
William Yorick likes assets. And 0 cost assets that he can easily get into the discard and have as an option for his ability regardless of current finances or available slots? That's pretty tempting. Plus, do to his recursion capabilities, Yorick isn't faced with the standard conundrum of when to brake his piggy bank - because he can do so in a free trigger window just before fighting an enemy that'll let him buy it back immediately.
Tommy Muldoon's signature weakness Rookie Mistake shuffles itself back into his deck if it doesn't trash anything with health or sanity on it. Because of its wording, Relentless is a rare asset that DOESN'T have health but CAN have damage on it. So while it'd never synergize with Tommy's ability, it IS vulnerable to his weakness. Depending on what you want Relentless for, this could be an awful downside, or a spicy bonus. I choose the latter because it isn't that hard to get into play and get at least one damage on it as bait for his weakness. Of course, the weakness hits ALL of his damaged/horror'd assets, so you still have to play around that, keeping your assets fresh, and getting them hurt/defeated in bursts, but that's how you wanted to play him anyways - keep one thing vulnerable and protect the rest. Relentless provides vulnerable bait that doesn't take up any of your valuable slots.
Last, but certainly not least, is Mark Harrigan - the Guardian's guardian. His special ability triggers whenever 'damage is placed on a card you control' - that's right! Dealing overkill damage 'places damage' on Relentless and triggers it! This gives Mark two major benefits. First it gives him more opportunities to trigger his ability to draw cards, perhaps even more times per round if using tricks to deal damage fast outside the investigator phase (such as with Dynamite Blast. Second, it gives him triggers of his ability without HURTING him!!! Normally the downside to triggering his ability is he and his assets now have less health left for further punishment, but Relentless actually has inherent BENEFITS for taking more and more damage! As if Mark wasn't already a great contender for Shotgun builds, between Practice Makes Perfect and Overpower granting a lot more card advantage and reliable boosts. Relentless supports with even more refill, and turning wasted overkill into BOTH cards and resources.
And I can't talk about weird rules consequences of this card's wording without taking a moment to acknowledge Solemn Vow. If you... really wanted to... you could use someone else's Solemn Vow to move damage OFF of your Relentless and put it on one of their assets with health. Outside of niche situations of killing off a doomed Mystic ally when nobody else is hurt, I haven't thought of a good use case for that. Note that you cannot use Solemn Vow to move damage TO Relentless because it doesn't have health and thus can't have damage put on it (except by its own ability which explicitly says to do so and overrides the general rule via The Golden Rule) the same way that Archaic Glyphs can put secrets on itself with its own ability but can't have them added via other cards.
Edit: Oh, I guess I missed that you can also save the damage to give it back later if you somehow haven't taken any yourself...