What is the way to get from Reykjavik to the Blue Lagoon for the cheapest price?
There is no local bus service to the Blue Lagoon so your formal options are essentially walk (the whole way could take days), hire a car or take a taxi. Assuming a ~70 km trek does not suit, there are unofficial alternatives that might suit.
There is a bus that runs between Reykjavik and the international airport at Keflavik. Cost for a singe (adult) is around 2.500 ISK (about $20). It does stop between termini but not where would suit you, so you would have to persuade the driver to be helpful. That might not be possible if a bus is running late, since it has flights to catch, but the service is roughly hourly and if the first won't co-operate, the second might. (The schedule is too frequent to be covered by the same driver on consecutive journeys.)
If to drop you off at the 43/41 junction just after Vogar you would be left with a walk of about five miles. However the second part of 'unofficial' is that Icelanders are very helpful to hitchhikers and, depending upon the time of day, there is a good chance of a lift for at least part of the way.
The cheapest option I'm aware of (short of hitchhiking) would be the buses operated by Reykjavik Excursions. You can buy a round trip ticket that includes admission to the lagoon for 9600 ISK.
Grayline also offers this service for 86 EUR (approx 10600 ISK).
In case you have a rental car, the admission alone is 5400 ISK. Do keep in mind that pre-booking is required.