Random name picker for a contest
You are correct. Math.random is indeed pseudo-random. However, its algorithm is actually quite good. In this SO post, a concern is brought up that the randomly generated numbers tend to all have the same number of digits. While this was statistically proven to make sense, you won't run into a similar problem since you are only dealing with single digit numbers.

Math.random should be random enough for your purposes. However, if you want to get truly random numbers to be absolutely fair, check out RANDOM.ORG. This, however, would obviously be a lot more unwieldy than just using Math.random.