The May 1945 meeting of the Target Committee is instructive for understanding the choice of targets.

They wanted targets that fit the following criteria: "(1) they be important targets in a large urban area of more than three miles in diameter, (2) they be capable of being damaged effectively by a blast, and (3) they are unlikely to be attacked by next August."

"Large urban area" means cities. They also wanted targets with military or industrial facilities in them, to better justify the attacks.

They ended up deciding on Kyoto, Hiroshima, Yokohama, and Kokura. They considered the Emperor's palace but decided against it (if you kill the Emperor, surrender would be difficult; by that point, as well, Tokyo had already been heavily bombed). Kyoto was removed from the list at the impetus of the Secretary of War, ostensibly because it was too "civilian" in nature (the Army disagreed with this assessment). Yokohama was removed because it was slated to be firebombed soon after the meeting. That left Hiroshima and Kokura. They added Niigata to the list before Kyoto was removed, but once Kyoto was off, Niigata was too far from the other targets to be a plausible primary or secondary target.

They added Nagasaki to the list right around when it was finalized in late July, to replace Kyoto. It had some war industries in it, but was a much more lower-priority target. Why? Because it had already been conventionally bombed several times (it was not a "virgin" target), and because its geography made it not as effective a target from the perspective of blast damage.

Hiroshima was in many ways an ideal target from the targeting committee position: it had a large army base in the middle of it, it was almost exactly the right size for a weapon of the 15-20 kiloton blast range, it had a lot of rivers which made it less ideal for conventional incendiary bombings, and it was geographically a flat "bowl" surrounded by hills, so the blast wave would be reflected back into the "bowl" to great effect. The operation was very carefully planned out.

Nagasaki is a strict contrast. The geography is poor — it is a long, snaking city in a valley, broken into two distinct parts. The atomic bomb could only really hit one of those parts, so the city was still functional the day after the attack (unlike Hiroshima) and there was no mass firestorm. It had been bombed before, as noted. It had some war industries but not nearly as many at Hiroshima. It was never the primary target: it was the secondary target for the second bomb, with the primary target being Kokura. Kokura was more like Hiroshima: a big, unambiguously military facility (an arsenal) in the center of the city, surrounded by workers' houses, and the city itself was the right size for the bombs in question. The Kokura bombing had been studied in great detail and planned out. Nagasaki was much more of an afterthought. The Nagasaki mission itself was somewhat of a cock-up on many fronts.

Anyway. The "strategic value" of these cities is not really the right question to ask. Did they have some military value? Sure. That by itself doesn't tell you why they were made targets, or why cities in particular were chosen as the first targets (as opposed to targets that would minimize civilian casualties). Which is to say: one might want to question the logic (and many did even during World War II — the idea of "total war" was always controversial and should not be taken as a simple explanation) of whether living in a city with a legitimate military target makes all civilians automatically legitimate military targets.

As for "alternative choice" — there were several "alternatives" on the table at the time. I have cataloged some of them here. The spectrum of alternatives, as I've tried to map out, are more than just "bomb vs. don't bomb" (or "bomb vs. invade") — they made a lot of choices about how they would bomb, for example.

This comment on one of the AMA's on the Manhattan Project deals with the choosing of targets.

The answer given deals with Yokohama and Kyoto but is applicable to most cities in Japan

Yokohama was already destroyed by firebombing by the time the atomic tests were ready. They wanted "virgin" targets that would showcase the power of the bomb, both to the Japanese and to the rest of the world. So Yokohama wouldn't do that.

Kyoto is such a trickier case. It was removed entirely on the initiative of the Secretary of War, Henry Stimson. The military actually pushed him pretty hard to get it back on the list, but Stimson rebuffed them and even got Truman to sign on to its being exempt from bombing. Stimson's official answer was that Kyoto was a cultural center with no military relevance, a purely "civilian" target, and that destroying it with an atomic bomb would make it much more difficult for the Japanese to be compliant with American leadership during the Occupation. His personal answer may have been related to the fact that he spent time in Kyoto when he was governor of the Philippines and loved it as a city. The military disagreed with him, as it happens, on the lack of military relevance of Kyoto — it had airplane producing factories and other industries, and it was a major transportation hub for materials (their proposed "ground zero" was the Kyoto roundhouse, which is now a locomotive museum). Anyway, it is a very curious case, and a nice example of one of the places where the idiosyncratic will of an individual can change the course of history, or at least its detail. I have written about Kyoto at some length here, including what I think its importance is for understanding Truman's apparent confusion about the fact that the atomic bomb victims were mostly civilian (I think Stimson's framing of Kyoto vs. Hiroshima as a "civilian" vs. "military" target is somewhat to blame).

The US had already launched incendiary attacks on 67 Japanese cities before they started using atomic bombs. Destroying cities from the air with fire clearly wasn't, by itself, shocking enough to get capitulation. On the other hand, if you are asking, do I think that aerial bombing and blockading might have produced an end to the war without invasion, it is not improbable (and the US Strategic Bombing Survey concluded as much in 1946). The Japanese were running out of resources; their options were either to try and negotiate surrender or to be suicidal about it. Not everyone at the top was in favor of the suicidal approach (though some were).

Edit: The answer was provided by /u/restricteddata