Lynn: Not exactly a "sword". It was a combination of a spear & battle-axe used in the 1500s & 1600s. See 'halberd' in a dictionary.
<Late Middle English 'haubert', 'halbert' <Old French 'hallebarde'<Middle High German 'helmbarte', < 'helm' "handle", "staff" + 'barte' "an axe", akin to 'bart' "beard". [for the sense, compare English 'barb' meaning both "beard" & "fish-hook"]. This is the view taken by three dictionaries that I consulted.
However, Middle High German also had 'helm' = "helmet", so a 'halberd' co-incidentally was a "helmet-splitting axe". This is Eric Partridge's view.
This would suggest that the Halbert surname was originally applied to halberd-bearers. I haven't see a Halbert coat-of-arms, but it would be interesting if it showed a halberd on it. These halberd-bearers may have functioned as royal or noble guardsmen.