In the context of this question, "money" is defined as the Pokemon Dollar. Other forms of currency like Battle Points will not be looked at. This answer will address items, moves, and Abilities that boost the money received from a given action (e.g. battling, selling). Unless stated otherwise, all mechanics mentioned will refer to Generation IX.
Abilities
As of the Indigo Disk, there are no Abilities that boost the money the player receives. Pickup is the closest to this: many of the items it generates can be sold for a modest price.
Moves
G-Max Gold Rush: Scatters coins that are picked up after battle. The number of coins is equal to 100x the user's level on the first turn of Dynamax, 200x on the second, and 300x on the third. A maximum of $99,999 may be collected per battle.
Happy Hour: Doubles the prize money collected at the end of the battle, including coins scattered on the ground. It will not have a greater effect if used multiple times in one battle.
Make It Rain: Scatters coins for each target hit: $5 multiplied by the user's level. There is no cap on money collected through this move.
Pay Day: Scatters coins 5x the user's current level. It shares the collection limit with G-Max Gold Rush in Sword and Shield. Coins are only collected if the player knocks out all opponents or uses a phazing move to end the battle.
Held Items
Amulet Coin: If the holder takes part in a battle, the total prize money received is doubled. No other factors (e.g. Klutz, removing the item, being knocked out) will prevent this. It cannot stack with itself or the Luck Incense.
Luck Incense: Refer to the Amulet Coin. It cannot stack with itself or the Amulet Coin.
Other Items
Prize Money Power: A Pass Power and O-Power obtainable in Generations V and VI respectively. It increases prize money by different increments: 1.5x at Level 1, 2x at Level 2, and 3x at Levels 3, S, and MAX.
Roto Prize Money: Triples the prize money received.
Stacks
Many of these methods will stack with each other. A list of these is given below.
G-Max Gold Rush: Shares a counter with Pay Day and stacks with either held item.
Happy Hour: Will double money made from Pay Day and Make It Rain. It stacks with either held item and the Prize Money Power.
Make It Rain: The scattered coins are doubled if any participant held an Amulet Coin.
Pay Day: Shares a counter with G-Max Gold Rush and stacks with either held item. It is boosted by Happy Hour and the Prize Money Power.
Amulet Coin: Doubles the number of coins picked up from Pay Day, G-Max Gold Rush, and Make It Rain. It stacks with Happy Hour and Prize Money Power, but not itself or Luck Incense.
Luck Incense: Doubles the number of coins picked up from Pay Day and G-Max Gold Rush. It stacks with Happy Hour and Prize Money Power, but not itself or Amulet Coin.
Prize Money Power: Stacks with Happy Hour and either of the held items. It multiplies Pay Day's coin output.
Roto Prize Money: The mechanics are still unclear, and it cannot be said with certainty that it stacks with anything.
Additional Resources
The following questions address effective ways to make money in many of the games. If this answer didn't fully meet your needs, please try one of the these instead.
How to make more money in Pokemon Ruby?
What is the best way to make money in FireRed/LeafGreen?
What is the quickest way to get money in Pokemon Emerald?
Where is a good place to earn money quickly in Diamond and Pearl?
What is the best way to get money in Pokemon Platinum?
How do I get lots of money in Pokemon HeartGold?
Where is a good place to get money in SS / Johto?
What are some easy ways to make money fast in White 2?
Fastest way to get tons of money in X/Y?
What are some easy ways to make money in ORAS?
How do I get money in Pokken Tournament?
How can I get more money in Sun and Moon?
How do I gain LOTS of money in US/UM?
What's the best way to get money in Let's Go?
What are efficient ways to make money in SwSh?
What is the fastest/most effective way to gain Pokédollars in BDSP?
