This is actually a really good question, and it has provoked some fine answers. I'm adding this because I wondered about this too, and believe that Monte Carlo was used and became popular, because the process employed by gambling casinos, and the statistical process of estimation have specific, similar characteristics. The methods employ randomness, but themselves are not random, as scientists typically use the term. (Eg. "I expected to see some evidence of xxxxxx, but the results looks completely random."). Both the Monte Carlo casino operators, and those using statistical techniques are seeking a specific outcome, and they are using similar methods to achieve a desired outcome. Random typically implies no evident pattern or unpredictable.
The methods used by gambling casinos are very well thought out. The unpredictability of the specific outcome of specific events is established (else it would not be gambling, would it?), but the nature and distribution of the range of outcomes is fully understood - and this key fact, both in casino gambling, and in the use of statistical estimation techniques - makes all the difference.
An example: A Monte Carlo roulette wheel will have 1 to 18 numbers in one colour, and 19 to 36 in another, if I remember correctly, so red or black have equal probability of appearing. You can wager a specific number, or just bet on red or black appearing. The players are playing against each other, for each other's money. How does the casino operator make any money from this?
The wheel has a "0" position, and if and when the ball lands there, the casino operator rakes in all the bets - the house wins. Each time the ball lands on the zero, the house wins. So each trial - each spin of the wheel - has a random outcome, but the house has (assuming the wheel is not rigged, say by putting a little magnet under the "0" number), then the house still can expect to - on average - sweep all the bets off the table with a 1/37 (or 0.027027) probability. And if the house wants to improve its outcome? It can add a second number to the wheel - typically "00", or double-zero. Now, the probability that the house will win is almost (but not quite) doubled, to 2/38 (or 0.0526316). That's over 5%, or a serious take. Suppose the average amount of money wagered at the table each night, by the high-rollers, is $ 170,000. With one zero, the house can be expected to make 170000 * 0.027027 = $4594.59, but adding the extra zero, and the expected take for the house is now 170000 * 0.0526316 = $8947.37.
See, the amount gambled each night will be random. We won't know what it will be. But assuming the wheel is fair and true (and smart gamblers are always watching to see if a game is "rigged" - just like the house detectives are always watching to see if players are cheating), we can say with close to certainty, that by adding the extra zero to the wheel, the casino can almost double it's take from the roulette game. If the casino operators can add that extra zero to the wheel, and not drive away players, and reduce the nightly amount bet, then they will do it. And just using simple probability, the improvement in the cash-take can be predicted. If the casino operator adds free drinks, to attract more players, then the cost of such extra attractions can be subtracted from the expected improved take. It may well be that adding the extra double-zero to the wheel and adding free alcohol, may improve the bet-flow. As the casino operator, you would run some experiments, and assess the outcomes. And since the operation of the process is filling your pocket with money, you are willing to focus on how it works, with some serious attention to detail.
And that, lastly is the key point. Although randomness is employed, the operation of any casino is a very analysis-intensive business, where statistical techniques and an understanding of probability are key to obtaining a successful outcome. Casino operators know the expected outcome on each and every game they offer, and because the randomness is limited to activity within a known distribution of possible results, the expected cash take on each game can be estimated quite accurately. This is how cheaters are caught. One particular game experiences a big divergence from expected outcome, right? As the casino operator, you know something is wrong.
Monte Carlo methods, or the techniques of statistical and probabilistic estimation, can be very effective at predicting the outcomes of processes where the distribution of possible results is known. And if either side can obtain an "edge", or shift the distribution of random outcomes in such a way as to alter the long-term expected value of the outcome even slightly, such an "edge" can make a person (or more often, the casino operator), very rich.
Monte Carlo methods employ randomness, but the methods - and the outcomes they can provide - are not random at all. The methods themselves can be as well-engineered and finely-tuned as the engines of the Porsche automobiles in the casino parking lots. And that is why the term is used.