Cannot remove part of $PATH
From your question it sounds like the problem is a directory ending in a /. And it sounds like Homebrew is telling you the directory causing the error is /usr/local/Cellar/node/5.0.0/bin/. If this is the case then you simply need to remove the trailing / from this directory.

But it also sounds like you are saying you don't see this problem directory in your $PATH. Running echo $PATH will show you which directories are in your PATH. If your PATH contains a lot of directories as mine does, this can be a little hard to read. So maybe you are just overlooking it?

In any case, here is a simple script to show you all the directories in your PATH; 1 per line. 

#!/bin/bash

# save IFS to a variable    
old_IFS=${IFS$' \t\n'}

#set IFS to a colon
IFS=':'

for item in ${PATH[@]}
do 
    echo $item
done

# set IFS back to the default setting
IFS=$old_IFS


Or if you prefer copy and paste directly in your terminal:

old_IFS=${IFS-$' \t\n'}; IFS=':'; for item in ${PATH[@]}; do echo $item; done; IFS=$old_IFS


Or an even simpler way as pointed out by fd0 in the comments:

tr -s ':' '\n' &lt;&lt;&lt;"$PATH"


This also has the added advantage of not needing to mess around with IFS.

Using any of the above scripts, if you find a line ending in a / then you've found the culprit.