What is “beer money”?

Beer money is the amount of cash set aside by the male head of the family, in order to spend a night or two out drinking with friends in pubs. The "beer money" in your excerpt, is talking about the amount of money earned by the entertainer, similar to tips, handed out by the regulars who attended the local.

I imagine that the entertainer is either a comedian who performed in small comedy clubs, notorious venues for heavy drinking and introducing new acts, or that he is a musician; perhaps a singer-songwriter and guitarist. They too perform in pubs in order to attract more customers.

EDIT: The entertainer however did not start his career by building up slowly over the years but rather he defines his (it could be a woman but I very much doubt it) as being an "overnight sensation" i.e instantaneous recognition and fame.

The longman dictionary defines "beer money" as:

beer money [uncountable] British English, informal a little extra money to buy a drink or have fun with: "The job was never going to make me rich, but it kept me in beer money for a while."

People often ask me to describe in detail how I've got to be where I am today, thinking that I've spent years earning beer money in local clubs...

As others have said, "beer money" simply means a small amount of disposable income. It's not unlike pin money, although I believe that expression is falling out of fashion.

That said, in this quote, I wonder if "beer money" (which was "earned in local clubs") is meant to imply, "I'd make just enough money performing my act that I could buy a beer before I went home" – and that the price of the beer wasn't all that much less than the take-home pay for the work. In other words, it would have been roughly a break-even endeavor.

That's just a supposition, but, if true, the quote would be akin to when someone talks about a job with low pay but high travel expenses, so that the pay barely "covers the gas money." As one person wrote:

You have to sell a lot to get a 50% discount and I can't see how that covers the gas money to deliver all that stuff...

It's an expression used to describe small amounts of money as set out in the answers above, and can sometimes be related to part-time earnings, or sums of money earned or raised in addition to a salary. I believe the term is also related to the historical or archaic notion of small beer, which was beer of a lower alcoholic content specifically brewed for children as an alternative to the drinking water which carried dangerous diseases.

small beer noun 1 chiefly Brit. a thing that is considered unimportant : even with $10,000 to invest, you are still small beer for most stockbrokers. 2 archaic weak beer.