Heirlooms could be just like Coppers early in the game but different later. For example
+1$
Count your discard pile. If there are at least 15 cards in it, +1$
or
+1$
+1$ per unique cost among Action cards you have in play
or
+1$
+1$ for each Province missing from the supply
The first one seems most reasonable albeit possibly annoying to play with, the second is probably way too strong and swingy since it might be worth 3$ on turn 3 or might be worth 1$. The third one needs a better wording but +9/-1 per Province in the supply doesn't work for multiplayer games; with that kind of wording you would first have to differentiate between the number of players in the game, then give out coins accordingly, then deduct coins for Provinces. These examples are more to illustrate the general directions in which I think Heirlooms might be going than real guesses.
Like Shelters, these designs only rarely change the opening, which means you don't get T1/T2 blowouts like you would with some of the designs suggested here. They might not all be like this, these are sort of the Necropolises among Heirlooms. If the trashing theme returns, other Heirlooms could be more like Overgrown Estate and Hovel (could even have the exact same text).
As to the question of how many Heirlooms there will be, the set has 500 cards and 33 Kingdom piles, with 0-1 Victory piles that would be 363-365 cards. The remaining 135-137 could be:
- Boons
- Hexes
- Travellers
- Events
- Heirlooms
If Fate and Doom are card types, this could mean that Boons and Hexes are in the supply, just like Looters put Ruins in the supply but cards adding Spoils to the game do not have an extra type. Hopefully there aren't 60 of each, otherwise there won't be much room for anything else.
I'm mentioning Travellers because this would be in line with Dark Ages' upgrading theme (Mercenary/Madman). Two Traveller lines would add 40 cards.
Given that Donald X said he's pretty much out of ideas when it comes to Events, most of the Empires events relied on the tokens in that set and that Nocturne will not have any tokens, there may or may not be some Events but there probably won't be a lot.
If we assume 0 Victory piles, 30 each for Boons/Hexes, 2 Traveller lines and 0 Events, that leaves us just 2 cards short of being able to fit in 42 Heirlooms, i.e. 7 per player for 6 players. If there are just upgrade piles à la Merc/Madman (or no upgrading) there would be enough room for all the Heirlooms. Which means they might replace all your Coppers, although this would of course make Moneylender/Miser quite sad, sadder even than Baron with Shelters. I'm leaning towards not all Coppers being replaced.