Where's the cheapest place in the world for a beginner to learn diving?

The answer is almost certainly Thailand, in particular Ko Tao, the PADI factory of the world. An end-to-end course that gets you the standard PADI Open Water license will set you back roughly 9800 baht (~$310 at time of writing); if you're happy with the far more limited PADI Scuba Diver course, you can complete that for 7000 baht (~$225) at this randomly chosen dive shop. Chuck in food and accommodation for around 1000 baht/day all-in for 3-4 days (you can squeeze by for less if you're a cheapskate frugal), and you can complete your course for under $500.

A close second is Malaysia, where diving is about the same price due to furious competition, but food and accommodation are generally pricier. I'd imagine the Philippines are in roughly the same price bracket as well. Diving in Indonesia is generally not as affordable simply because the diving off Bali is not that great and the places with great diving (Bunaken etc) are in harder-to-reach locations.

All that said, diving is one of those things where you definitely don't want to look for price alone, this is life support equipment we're talking about here and you want to make sure it's properly maintained. (When I lived in Singapore, I ended up buying my own gear, just so I could maintain it myself.) So do your homework and pay a little extra to find a dive shop with a clue, here's a handy article with some tips. (TL;DR: Check online reviews, go visit, chat with the instructors and look for certifications, general cleanliness, gear appearance and maintenance schedule.)

Alicante, Spain. You can get some offers from €130 for a two-day course. Normal prices are a bit higher (€200), but if you just want to get the PADI certificate, this is the cheapest way I've found. In Alicante most people speak fluent english and sleeping and eating each day is about €25 if you go to a hostel or similar and cook your own food. Also, if you live in Europe or even in America, the flight is much cheaper to Alicante than to any south Asian country. The sea is the Mediterranean, not so impressive as the Thailand seas, but not too bad.

Lake Malawi, Malawi it's $350 for the open water padi cert at aquanuts. In terms of accommodation next to the dive shop is Kande Beach, if you have your own tent $5 a night otherwise it's a cabin for $15 per person per night. The food is cheap and so is the beer.

So unless you bring a tent you are looking at $500.

Cheaper Is Not the Way

For the purpose of safety I will start off by repeating what others have already mentioned: looking for the cheapest place to learn something potentially life-threatening is probably not a good idea. Ask yourself how these places can afford to charge less for the organisation of an activity, and the maintenance of gear, both of which are inherently expensive. If costs are reduced by cutting the materials and maintenance budget, then you'd probably not want to be trained in such a club.

Go Local with Group-Buys

When I first qualified in 2004, the average price for a first scuba diving certification, be it PADI or CMAS, oscillated between 300€ and 400€, back home in Italy. This included renting gear, learning materials, pool training and two or three sea dives (excluding accommodation). However, the recent boom of deal-of-the-day, group-buys and other similar websites has reduced prices of such activities dramatically. You will most probably find a local dive club offering cheap rates for open water certifications on Groupon, for example. Here is a randomly-picked offer at 125$ for Open-Water PADI. Beware the link will inevitably expire soon, but you can do more research yourself.

The main advantage here is that you cut out travel and accommodation costs, since you wont be flying to the Pacific in order to take the course. Obviously, unless you live in places where the seaside offers amazing flora, fauna and landscape, you might have to settle for cold-water dives in quarries (yes UK residents, I am talking to you!). However I would say that this is what you get when looking for the cheapest deal.

South Africa offers Open Water Diver courses in Range of R2000 - R6000 ($200 -$600). Depending on training organisation, location, time of year, additional options, etc.