Investing in the stock market early is a good thing. However, it does have a learning curve, and that curve can, and eventually will, cost you.
One basic rule in investing is that risk and reward are proportional. The greater the reward, the higher the risk that you either (a) won't get the reward, or (b) lose your money instead. Given that, don't invest money you can't afford to lose (you mentioned you're on a student budget).
If you want to start with short but sercure investments, try finding a high-interest savings account or CD. For example, the bank I use has an offer where the first $500 in your account gets ~6% interest - certainly not bad if you only put $500 in the account. Unfortunately, most banks are offering a pittance for savings rates or CDs.
If you're willing to take more risk, you could certainly put money into the stock market. Before you do, I would recommend spending some time learning about how the stock market works, it's flows and ebbs, and how stock valuations work. Don't buy a stock because you hear about it a lot; understand why that stock is being valued as such. Also consider buying index funds (such as SPY) which is like a stock but tracks an entire index. That way if a specific company suddenly drops, you won't be nearly as affected. On the flip side, if only 1 company goes up, but the market goes down, you'll miss out. But consider the odds of having picked that 1 company.