Select the word around the given index in a given string
V, 10, 9 7 bytes
À|diwVp

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This answer uses 1-based indexing.
This could be shorter if we do exactly what the title says: &quot;Select the word around the given index in a string&quot;. We could do
À|viw

Which literally selects the word, but unfortunately doesn't change the output at all. So we need a little workaround to make it work by cutting it into a register, deleting the rest of the text, then pasting the register back in.
Explanation:
À|          &quot; Jump the position of argument 1
  diw       &quot; (d)elete (i)nside this (w)ord.
     V      &quot; Select this line
      p     &quot; And replace it with the word we just deleted