Well, without seeing the sheet music, I think this is how I would start...
First, by looking at the notes themselves, and seeing how they're grouped. Could we assign chords to them? If so, where do the chords switch? If there is a consistent pattern, that may help identify the time signature.
Second, I know that in general notes will fall on the downbeat of a count. So counting from the first note, if I see quarter note, 8th, 8th, quarter, quarter, 8th, 8th, etc, then I would note that counting by eighths I have beat pause beat beat beat pause beat pause beat beat. Right away I see that there is a beat every other note, a pattern that continues onwards. Might it be in 4/4 then, or 6/8? Some even time signature, certainly. What if that breaks up and I have a quarter note's second eighth (if that makes sense) where an eigth note should be? Could be an odd time signature then.
In a similar vein, how about counting the timing of the bass notes? Bass notes are more likely to occur on downbeats, ie, to provide a sense of rhythm and emphasis. When they are present, that would definitely be an indicator to me of time signature.
To explain this a different way, look at the series of notes as a group of 5 eighth notes. Does it make sense? Is there a note every 5 eighths? Could be 5/8. Is there a note every 6, or 7, or 8 eighths? etc.
I've included a few links to different odd time signature resources, maybe that might help you.
Saul