I believe these are not very old names, and the meaning might depend on the language and Indian spelling. The -esh bit of the name actually comes from Isha which literally means owner or possessor, though master, both literally and figuratively, is often a better translation. It commonly formed the second part of two-part words meaning `master of' (e.g. vAgIsha, master of speech, ramesha, husband of the enchanting ramA, etc.) I think hitesh and ritesh are modern coinages in the same mold, but I do not know for sure.
hita comes from a very old word whose root meaning is `put', `established', or `arranged', but the only meaning that survived into the later languages was `good' or `beneficial'.
rita (and here I am guessing which of the many different roots it might come from ... the spelling in any script of Brahmi origin would have helped) is also a very old word which is from a root meaning to go or rise, and actually expressed the qualities of easy going straightness. Figuratively it stood for straightforwardness and honesty (someone who follows the way), qualities that made a person Arya, or honourable. (It is this latter word which was appropriated to extremely unfortunate use in the 20th century.)
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