How to avoid ambiguity when referring to a 24-hour period?

There's nothing weird about saying 24 hours, but if you want to avoid it you can say until / at / by this time tomorrow.

First off, make up your mind if you are looking at day or 24 hours — they are not quite the same thing! The first, Nychthemeron, is I feel not very relevant here.

Rather, the OP means a period of 24 hours from a reference point.

Considering this, (and in order not to sound weird), I'd actually just say, "Same time, tomorrow."

It depends on how exact you need to be.

• If you mean precisely 24 hours (for instance, for business purposes - some banks freeze funds for exactly 24 or 48 hours after a deposit), saying "24 hours from now" is fine. It also works with 48 hours and occasionally 72, but after that you're getting unwieldy and you're better off just saying "next Tuesday at 3:57."
• If you're a little looser in your requirements (i.e. asking a friend to meet you at around 2pm tomorrow when it's 2pm today, but not assuming that he'll be there on time to the second), then something along the lines of Kris's answer is best.
• If you're being any vaguer than that, just go with "tomorrow."

Nychthemeron - n. The natural day and night, or space of twenty-four hours.
Webster's Revised Unabridged Dictionary, published 1913 by C. & G. Merriam Co.