Yes, suppose could be water in the sample though CDCl3 is usually pretty dry and surprised you'd be given a problem like this with a confusing peak like that.
Apart from this, does fit with 2-bromobenzyl alcohol as you suggest and the m.p. is spot on!
You might be along the right lines for 1,2-disubstitution but integration of the 2 singlets are the wrong way round for this. CH2 of a benzyl alcohol should be ~4.7 but only integrates to one proton.