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So I was analyzing a song on the piano and saw a broken chord with notes B, F# and A (all ascending). Now it looks like root note, perfect fifth and minor seventh. If you add minor third, it becomes Bm7 chord. But is there a name for a 7th chord with the third missing? I looked around on the internet, couldn't find any.

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You don't say what the instrumentation is, but it's important to recognize that determining the chord at any given point needs to consider all the parts that are playing. In particular, the 3rds of chords are very often only played in one part, and not doubled by other parts. If I saw such a broken chord, I'd assume that there was either a D or a D# in another part -- perhaps in a vocal lead, or in a bass line -- and that the chord was indeed a B7 or Bm7. Indeed, skipping the third in an arppegiated seventh chord is fairly common for this very reason.

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More information is needed for a useful analysis. With the missing context, we might label it as either a Bmi7 or B7 without worrying about the missing note. Without any of that information we can formally say that it could function as either a Bmi7 or B7. To narrow it down further would require more information.

One would not explain it away as a different name like we do power chords (chords with the root and fifth Exclusively), unless it is in fact in a context where there is mostly power chords used. In that case perhaps we could give it a new identity such as B57, but such a name is not in common use since I just made it up (Perhaps Heinz ketchup should use it in their jingle). Chuck Berry would play power chords like this B5(B-F#), B"6"(B-G#) B"7"(B-A) where the B6 and B7 are missing all but the root and the 6th or 7th respectively. It would not be a problem to add the 5th to the B7 and have it still sound good. I have never heard of these chords being described as having a special name because of the missing notes. This is because, (opinion warning), adding the notes back in would not ruin the sound in any way except possibly voice leading. Power chords are a useful short hand for the value of chords because in certain styles the performer does not care what other notes would fit. One size fits all, so further distinction would be a distraction.

As a side note, while any note can go missing from a chord and still be called the same name it will be more common (outside of power chord land) for the 5th to go missing. In fact, performers often remove the 5th automatically is a common feature (with many exceptions) in jazz.

It is possible that some special name for the chord could be coined if there were a style where such a configuration is used more often, similar to Punk's relationship to power chords.

I was once called out in a theory class for assuming that the root could not go missing too, when in fact it can. Perhaps it is implied or in a different part. It is also important how the chord sounds in context. Does it sound like the same chord in context, even with the root missing?

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