The regenerating heads are the whole design of the episode. Cutting is the obvious response and it makes the problem worse, so the solution has to be a change of method rather than more effort. Heracles has his nephew Iolaus sear each stump with a brand as it is cut, and the heads stop returning. The immortal central head is buried under a rock, because it cannot be killed at all and can only be put somewhere.
Eurystheus refused to count the labour. Heracles had accepted help, and the terms did not allow it, so a second labour was set. That refusal is worth noticing: the story goes out of its way to say that the clever solution was against the rules, and the cycle is longer because of it. A crab sent by Hera to bite Heracles during the fight appears in the same passage and is put in the sky as Cancer, which is a second reason the episode stayed in circulation.
The name survived the story. Linnaeus gave the genus Hydra to a small freshwater animal that regrows from fragments, so the classical monster is now a laboratory model for regeneration. Political writing of the early modern period used the hydra for any enemy that multiplies when attacked, and that metaphor is still the most common use of the word in English.