I imagine there's a connection between:

present - the time between past and future, and

present - here; not absent.

Is there? Are these related to present - a gift?

They're all of them, via Old French, from the Latin present participle præsens 'being before, being present' of præsum (præesse, præfui) 'to be before, present'. Cf. president from præsedens 'sitting before'.

And of course there's always preeZENT, which I assume is related, but isn't it interesting that the pronunciation changed?