what part of speech is "count" in "make it count"?

I guess it's a verb.

I'm trying to think why it means what it means. I guess it's related to 'counting' the score.

We also say that something 'doesn't count' if it's invalid for example if you broke the world record for running but no-one saw it and there was no proof - then it 'doesn't count'.

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From M-W Online:

Count

Etymology: Middle English, from Middle French conter, compter, from Latin computare, from com- + putare to consider.

Intransitive sense: to have value or significance.
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In this case, "making it count" means give it maximum effort.

That's like saying "make it matter", and so it appears "count" or "matter" used like that are verbs.