My understanding is that president Johnson mistakenly announced it as the SR-71 (versus RS-71) and thus the designation stuck (the emperor has no clothes)
The SR-71 designator is actually a continuation of the pre-1962 bomber series, which ended with the B-70 Valkyrie. Late in its career, the B-70 was proposed for the reconnaissance/strike role, with an RS-70 designation. The "RS" prefix (sometimes written as "R/S") was actually allowed as an explicit "special case" in the orignal 1962 issue of the designation regulations. When it was clear that Lockheed's A-12 aircraft (then used by the CIA) had much greater performance potential, it was decided to "push" a USAF version of that one instead of the RS-70. This USAF version was to become the RS-71.
"Conventional" wisdom now says that then president Lyndon B. Johnson messed up the designation in his public announcement and called it the SR-71 - and nobody wanted to correct the president. Because the strike mission had been cancelled anyway, "SR" was quickly reinterpreted as "Strategic Reconnaissance". However, a first-hand witness of those events recently revealed in Aviation Week & Space Technology, that LBJ did not misread anything. In fact, then USAF Chief of Staff LeMay simply didn't like the "RS" designator - he already objected it when the RS-70 was discussed, preferring "SR-70". When the RS-71 was to be announced, he wanted to make sure it would be called SR-71 instead. He managed to have LBJ's speech script altered to show "SR-71" in all places. Using archived copies of LBJ's speech, it can actually be verified that it reads SR-71 both in the script and on the tape recording. However, the official transcript of the speech, created from the stenographic records and handed to the press afterwards, shows "RS-71" in three places. It seems that not the president but a stenographer did accidentally switch the letters, and thus create a famous aviation "urban legend".