I also once heard such a story, but I have doubts it is literally true. What is an established fact is that Banach had an unusual start of his career.
He was actually an engineering student (with a personal situation rather on the difficult end) and did math more or less as a hobby. By pure coincidence he met Hugo Steinhaus who was impressed. They worked together and published something together. Then Banach got a position at a university (Lvov) and then a doctorate (under Lomnicki)Lomnicki [correction: while he was working for/in the group of Lomnicki, it appears Lomnicki was in no sense the director of his thesis; cf Magaret Friedland's answer]). So he got his doctorate under somewhat unusual circumstances and not following standard rules (though at that time, there were much less rules for doctorates then nowadays anyway).
In that sense, it was likely not so clear when and how he should submit his thesis, and it seems very conceiveable that he discussed this matter with various people and/or people close to him pressured/encouraged/helped him to do so. (Added: I see Francesco Polizzi made a comment sort of in this direction.)
Regarding the "laziness":
Not long after the time of his thesis he wrote a lot (including high-school textbooks). So, to attributed this to sheer laziness in a classical sense seems certainly odd. If anything I could imagine a certain uncertainty (and/or occupation with other matters) regarding how to proceed; or how to really write mathematics (not being trained as a mathematician).
Yet, it is also well-cocumented that he and others worked a lot in cafés. Now, this could to some be taken as a sign of a 'lazy' life-style. But, well, not even this is so clear.
For an overview of Banach's life http://www-history.mcs.st-andrews.ac.uk/Biographies/Banach.html