I suspect the answer depends strongly on your initial resources and just what you're trying to achieve. Your best prospect for becoming a billionaire might not be the same as your best for becoming a millionaire. (Though, I dunno, it might.)
If you're young, intelligent and living in a fairly rich country, and don't mind waiting a while and doing some work in order to get the money, your best bet may be the unexciting one of trying to get a reasonably lucrative job -- software development, medicine, law, finance -- and then work hard and live frugally for (depending on other factors, notably including luck) somewhere between about 5 and about 20 years. Given sufficient determination, this gives you an excellent chance of becoming a (dollar) millionaire, and a quite respectable chance of doing much better than that. But it requires a lot more time and effort than buying a lottery ticket; this probably doesn't meet your "reasonably low barrier to entry" criterion.
Many gambling games have very little "friction". Some (e.g., poker) greatly reward intelligence and effort; if you can make yourself a good poker player then you can probably turn (say) $1000 into $1000000 with probability somewhat over 10^-3. Some (e.g., roulette) are basically pure chance, so no investment of time and effort is required or indeed helpful. If you want to play them at high stakes you'll probably need to go to casinos and put up with their rake-off, but the friction is much less than in lotteries.
Once you've turned your (presumably small) initial investment into something more substantial (with low probability, but let's not worry about your counterparts in other Everett branches who weren't lucky enough) you may do better to switch from pure gambling to stock market speculation. There's still friction, but I think brokers will eat less of your money than casinos.
I doubt there are many opportunities around that have genuinely low barrier to entry and let you do much better than pure chance would allow -- because if there were, they'd already have been used up by arbitrageurs.
I haven't been able to find the source of the idea, but I've recently been reminded of:
This is, of course, based on the Multiple Worlds Interpretation: if the lottery has one-in-a-million odds, then for every million timelines in which you buy a lottery ticket, in one timeline you'll win it. There's a certain amount of friction - it's not a perfect wealth transfer - based on the lottery's odds. But, looked at from this perspective, the question of "should I buy a lottery ticket?" seems like it might be slightly more complicated than "it's a tax on idiots".
But I'm reminded of my current .sig: "Then again, I could be wrong." And even if this is, in fact, a valid viewpoint, it brings up further questions, such as: how can the friction be minimized, and the efficiency of the transfer be maximized? Does deliberately introducing randomness at any point in the process ensure that at least some of your MWI-selves gain a benefit, as opposed to buying a ticket after the numbers have been chosen but before they've been revealed?
How interesting can this idea be made to be?