"Going Back to Houston" from where?
Dean Martin sang of being in some city where he was broke, wet, hungry, and unknown. He sang of going back to Houston, but what city was he in? Was it a real city where he had been? I can't find anything on the historical background of this song.
I know he didn't write the song, but sometimes songs can still be personal, as with Papa was a Rolling Stone by the Temptations...Dennis Edward's real father apparently did pass away on the 3rd of September.
As Dean Martin wasn't a songwriter, it would be unlikely it had any personal meaning for him, except in the self-contained world of the song itself.
The lyrics are from the perspective on a down on his luck drifter who describes himself as "A walkin' case of the blues." The verses describe his many misfortunes ("I found a dollar yesterday, but the wind blew it away"), each ending with the repeated refrain "Going back to Houston, Houston, Houston...". The mood lightens on the final verse when the narrator reveals that despite his woes, he has a girlfriend waiting for him in the titular city.