How Many Cards?
A hand of cards does a lot of good things for a game. It provides choices. It helps tie turns together, by allowing the player to formulate a longer-term plan.
Cards can also be on the table, in a kind of "market". However, these may be taken by other players, and they also add to information overload.
Cards in hand are also hidden information, which pushes your game away from calculability, and towards intuition.
It's harder to design games that don't have a hand of cards.
Most games should have a deck, from which the players draw a card each turn.
Lots of cards
There's a problem with medium-complexity games that revolve around a hand of cards: the hand is not big enough.
A typical-sized hand of 5-7 cards will contain cards for all kinds of situations. Many will not be relevant to what's going on in the game.
Even if they are all relevant, the player doesn't really choose which ones to play — they just play all of them.
To increase choice, you want the player to have vastly more cards than the amount they play. How do you do that?
Give the player far more cards than they need, but also provide some way to remove unwanted cards from the game.
Unwanted cards
In one of my prototypes, a few actions simply require that you discard one card. It's not much, but it makes every card useful.
Race for the Galaxy is an excellent game. In the game, you draw lots of cards, but you also discard cards as the resource to pay for other cards. This lets you see many cards per game. The downside is that you're regularly discarding your whole hand, rather than planning for the long term.
These days, I avoid "choose and discard cards", whether this is one card, many cards, or all cards above a hand limit. This discarding forces the player to make an unimportant yet complicated and slow choice.
In Star Wars: Unlimited, you draw two cards per round, but you choose one to leave face-down as a resource. This is "choose and discard a card", but the game is slower than other games, so this happens much less often.
In War of the Ring: The Card Game, every time you play a card, you must also discard one.
In Radlands, each card can be "junked" — you discard it for free to get a small effect of some sort. So, each card has two ways to use it. This has the problem that each card is extremely useful, so I have to give less of them, which can reduce choice.
In my duel game, you just draw a new hand each turn. There is no choice, and you have lots of options. This destroys the long-term planning in your hand, but the huge choice and ease of play make for interesting turns.