When it comes to dice games, there are several types. There are the quick filler games, like LCR and Liar's Dice, and longer strategy games like Kingsburg and Roll Through the Ages. Several dice games take a pre-existing property, and transform it into a dice version. Hasbro experimented with this idea for Battleship, Monopoly, Risk, Clue, and Sorry and called it their Express line. While I appreciate the attempt to make these big name games faster, it really changed too much of the game's identity. Two of the big names in hobby games did this as well, Catan and Carcassonne. Ticket to Ride incorporated dice into the game, replacing the train cards, but it is not solely a dice game.
Let's talk more in depth about a few of these games.
LCR is a very simple game. Each player starts with 3 chips. Each turn they roll 3 dice. For each L, they pass a chip to their left. For each R, the pass a chip to their right. For each C, they put a chip in the center. The goal is to be the player who has all the chips at one time, meaning no other player has chips in front of them. There isn't any strategy to the game, just roll the dice and do what it says. I can't recommend this for gamers, but it could be a fun party game, or maybe a drinking game.
In Liar's Dice, the goal is to be the last person with dice left. Each player starts with 5 dice. All players roll their dice, and then the start player makes a statement about all the dice at the table, based on only knowing their own. The next player can either call them a liar, in which case all dice are revealed, and then if the asserter was right, the player who called them a liar loses a die, but if the asserter was wrong, they lose a die. If someone doesn't call liar, they must up the statement in some way, either by calling more of the same die face, or by calling a higher die face. For instance, if someone calls four 3's, I could call four 4's(5's,6') or five(or more) any face.
Liar's Dice actually has some strategy and some bluffing, so it is a real game. The dice are crucial, but so is the ability to read people, and do some quick probabilities in your head.
Catan Dice Game
The Catan dice game is an interesting play. Each turn, you roll 6 dice up to 3 times. The faces of the dice are the 5 standard Catan resources and gold. Gold acts as a wild, which is a great help. The game only lasts for 15 rounds, which keeps play time under 30 minutes, maybe 45 as an absolute long. In the game, you use resources to build roads, settlements, cities and knights. Knights grant you one free resource of a given type once during the game. They don't offer many points, but they can be a swing in the game. Things have to be built in order, which is a little confusing at first, but it keeps the game moving along. I've only played this a couple of times, but I find it a short version of Catan without the trading, and a fairly good competition between the players. There's a lot less you can do to mitigate bad luck, but given the time length, that's ok.
Kingsburg fits more into the boardgame that heavily uses dice, but I think that it is a dice game at heart. In Kingsburg, players roll dice, then take turns placing one or more dice on various advisors. These advisors give the players different resources or army points, or victory points. Once an advisor is claimed in a round, no one else can take it. There's a lot of strategy in how you place your dice, but even the best strategy can be foiled by poor rolls. There is a lot going on here beyond the dice, but if you like dice games that let you do something meaningful with your rolls, Kingsburg is a great choice.
Roll Through the Ages
RTA is a civilization building game with dice. Players have to balance workers, food and coins with the threat of disaster. The more cities you build, the more dice you get to roll, but the more people you have to feed each turn. If you can't feed your people, you lose points, but if you don't have enough cities, you won't have enough dice to do things to score points. RTA has a fine balance to it, but there are a lot of ways to win. I think it's one of the most replayable dice games out there.
I wrote a full review of Quarriors, which you can read here. As such, I'll just summarize. Quarriors is a dice building game, where you are trying to gather creatures to score points. Each player starts with a very basic set of dice, but they buy more each round to grow stronger. Quarriors has a lot of fans and a lot of detractors. I personally fall in the "meh" category. It's a decent game, but I don't want to play it frequently.
Here's another game I've done a full review, which you can find here. Scary Tales is all about rolling sets of icons to gather relics. There are cards that add in symbols, making the set easier to obtain. There's not an abundance of strategy to the game, but it is a neat game. I recommend picking up two different sets for variety, and to allow more players. I didn't really like it as a 2, but it's fun with 3 or 4.
Since we are building on a budget, let's look at the costs of these games.
If you're on a really tight budget, I'd say salvage dice from other games and play Liar's Dice/LCR. If you've got a bit more money to spend, Roll Through the Ages is a really great value purchase. For kids, Scary Tales might be fun due to the theme.
My pick for a dice game to own is Roll Through the Ages. Word of advice, track down "The Bronze Age" expansion, it's available for free online. It really does add to the game.