It really depends on how complex you want it to be. Keep in mind, the more complex, the easier to break.
Interestingly enough, you can actually make a fairly believable stock market simply by flipping a coin. Heads it goes up, tails it goes down. There is a well known anecdote about a professor who did this and was able to trick an investor who relied on reading charts.
Another basic way is to have trends. Look at the last time it changed and skew the probability to have that same result again. This will create more upward and downward trends throughout the history of the price.
If you want to be able to modify it someway, just have a basic 50/50 split whether up or down, and then provide certain modifiers that will change the probability. I would add diminishing returns so the more modifiers pushing the price one way or another will result in less change.
Any of these methods can work as a total overall price, set up for individual stock prices, or groups of stock, say for industry or whatever commonalities they may have.