Which is correct: "Filename", "File Name" or "FileName"?

The original form of the word was "file name", as in the name of a file. These days (and probably for a good few years), the compound "filename" is widely accepted and perhaps most commonly used. Either is of course perfectly acceptable.

Do not, however, capitalise letters in the middle of a word, under any circumstances. (Unless you are writing variable names in code.)

Filename is in my experience the most common and in my opinion the best looking. File name is also acceptable, but I would only use it rarely, perhaps in a parallel construction such as the file name and size. I find word-medial capital letters distracting and unpleasant anywhere but program source code, so I would never even think of FileName. Edit: if that's the reason you're asking, in program source I would still use Filename for a class or filename for a variable, not FileName nor fileName, respectively.

The British National Corpus has 240 cites for filename and 72 cites for file name. It's not possible to search case-sensitively, but several reloads of the random sample of specific entries gave me no matches for FileName.