Franco offered to join the Axis and the war, but there were strings attached...

and military and financial support from the prospering German and Italian states could help solidify the shaky regime of the new nation, allowing them to expand into Portugal, southern France and North Africa, and take back Gibraltar from the British.

...and its funny that you mention this because these weren't too far from what the actual strings were. They wanted arms, food, supplies. They wanted Gibraltar from the British and certain French colonial holdings in Africa.

Spain was, as you note, fragile after the Civil War, both politically and economically, so if Franco was going to bring them into this conflict, there was a lot that he wanted in return. Hitler felt that the asking price was too high - and the territorial concessions would have involved damaging the relationship with Vichy France - and so serious negotiations ultimately broke down.

Instead, they walked away with a rough and vague agreement that Spain would join the war... sometime. And Germany would let them have... something... in Africa. Obviously this never materialized. Franco planned to enter, but only when the war was near its end (and still hoping for supplies and material from Germany), as not to further the strain on Spain. As Germany began to lose, this possibility was all but eliminated.

A lot of this could get into specifics about the meeting at Hendaye in October 1940 and it seems that there's a lot out there about what ended up being a rather brief event. This paper in particular was interesting to me because it attacks the assertions that: (1) Franco knew he was asking too much, (2) knew that he'd be refused, and (3) that it was all a strategic move to keep a damaged Spain out of the war and save face at the same time as being revisionist. These were ideas that, until reading the paper, I'd heard and seen discussed seriously so I really appreciated the fresh perspective.