I remember that my father insisted on calling them "Hartlepools" when we met them in the FA Cup during the early 1970s.
It appears that this stems in the main from the 1908 foundation of the present club. Although playing at the Victoria Ground, situated in West Hartlepool, the team was named Hartlepools United against public opinion as it was decided by the monkey hanging powers of the time ( that's another story) that the team should represent both of the two Hartlepool boroughs (Hartlepool and West Hartlepool) that existed at the time.
Some considerable time later, the team and the boroughs both became singular and so In 1968 the "s" and the "United" were dropped from the team name of "Hartlepools United". This was in connection with West Hartlepool being absorbed along with the old smaller town of Hartlepool and the village of Hart into one new borough named "Hartlepool". The appendage "United" was restored in 1977.
Thanks to [Wikipedia] and [When Saturday Comes: The Half Decent Football Book By Tim Bradford] for providing the information reproduced in part above.
Perhaps they could have thrown "Hartlepool Boroughs" or "Hartlepool Towns" into the mix as well.