Probably for one of two reasons, or both:
New York State, at the time, was the most populous state in the Union and therefore would benefit the Democratic column in terms of Electoral Votes.
Also, the Roosevelt name carried a LOT of weight in 1920 - probably the equivalent of the Clinton name today. He was a member of a popular family that could sway an election (i.e. sway the liberal, or progressive, vote to the Dems).
Hope that answers your question 
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Considering how much President Wilson pissed off the Irish Catholic and German voters in New York earlier that year it would seem quite self destructive to put Franklin Roosevelt, who at the time was a vocal critic of Tammany Hall, on the ticket. The Roosevelt name may have been influential, but the influence of Tammany Hall (and other urban machines) over certain demographic groups was a lot stronger. Wilson already screwed over his party in New York when he blamed German and Irish Americans for the lack of support for the League of Nations. Appealing to progressives won't do jack sh*t in New York in 1920 when you have already pissed off a significant chunk of your own voting base by blaming them for why (insert x name here) is unpopular. For every progressive that might've been swayed by the Democratic ticket there would've been 2 to 3 Irish Catholics staying home and 2 Germans voting Republican come election time.
Considering that the Democrats lost New York by seven points in the Election of 1916 (a somewhat better year for the Democrats), the idea that they would've attracted enough voters with FDR on the ticket to possibly win New York makes even less sense. I mean one of the reasons Wilson survived re-election was because he assured voting bases like German and Irish Americans that he would not get involved in World War I. The US joining the Allied side in World War I and the League of Nations were the closest thing to metaphorically pissing in those voting bases faces as one could get. Hell, if they did think like that I'm not surprised they got such an epic ass whooping.
My theory? The Democrats went out of their way to screw themselves in 1920. You want to screw up an election as much as possible, just do what these guys did. Franklin Roosevelt's last name may have been Roosevelt but he was still Franklin Roosevelt, a name that was not popular at all with the boys who ran Tammany Hall or their votes.
Sometimes the conventional thinking on this website just sucks.
