Aside from power chords, if you need to find the chord for a single note, you're also going to need to take a look at the other notes.
In the most general of cases, you're going to be able to derive the scale/key of the melody; e.g. if the melody contains a F# and a C, chances are that it's going to be in the key of G Major/E minor. The so called 'circle of fifths' is a great way to visualize that - look it up.
Once you've got the key, you can pick chords from that key that match the note. For example, in the key of G, the note of E could work with Em (E-G-B), C (C-E-G), Am (A-C-E), and maybe also F#7 (F#-A#-C#-E - which has two out-of-key notes right there).
F#7 doesn't belong to the key of G (it's a substitution for F#dim), but still works - which only serves to prove that there are many exceptions to this rule, and it is no more than a general method of figuring stuff out in some cases.