score:5
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.
Upvote:2
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.