Question: For instance playing standard barre on the 2nd fret, but leaving the high E open.

My brain is banking out figuring out what an E is in this context and how the resultant chord would be named.

Answer: 
This might be called a Bm(add11) or Bm(sus4), because if you're playing it like so:

```
7 9 9 9 7 7 

```

(which is a common barre form for adding the 4th above the chord's root), you won't have the third of the chord. On the other hand, if it's like this:

```
x 2 4 4 3 0

```

then like Dom says, it is a Bm(add11).
It's just a Bmadd11 since you're just adding a fourth(aka an 11th) to the chord.