Donald X recently floated the idea that Copper need not be a supply pile. This affects surprisingly few cards and would free up 32 cards worth of space in the base set (so 3 new cards for the base set using a blank).
Suggestions for cards that would need to be replaced or upgraded for functionality,, trying to retain the spirit of the card. In quotes where I think it's different enough to warrant a new name:
"Beggar"
Action/Reaction - $2
Gain a Silver to your hand. Put from one to three cards from your hand onto your deck for $1 per card topdecked.
_________
When another player plays an attack, you may play this.
"Treasure Trove"
Treasure - $5
Gain a Gold. You may discard a treasure to gain a Silver.
Count
Action - $5
Choose one: Discard 2 cards; Put a card onto your deck; Trash a card costing $3 or more
Choose one: +$3; Gain a Duchy; trash your hand
"Jester"
Action - $5
+$2
Each other player reveals the top card of their deck. You may gain a copy of a revealed non-Victory card. +$1 if you didn't gain a card.
Alliance
Event - $10
Gain a Province, Duchy, Estate, Gold, Silver and Curse
"Banquet"
Event - $3
Put 2 cards from your hand onto your deck to gain a non-Victory card costing up to $5
"Fountain"
Landmark
When scoring, worth 10VP if you have 6 or more coppers
"Greed"
Hex
Put a card costing $0 to $2 from the trash onto your deck
(Note: I don't think hexes will survive)
Mining Road and Barbarian come to mind as cards that would be meaningfully changed by not having Copper in the supply but I think they're fine as is.
Edit: Modified Greed