Reflexive possessive pronoun

Solution 1:

I would use the singular their for this purpose:

One of us will see the money in their account.

It's useful for purposes besides referring to someone of unspecified gender.

EDIT: Upon further consideration, my modification to Andrew Leach's answer is just as good (and better for people uncomfortable with the singular they in this context).

One of us will see the money in our own account.

To me, our suggests a joint account, but adding own removes that implication, and suggests a single-owner account. Our own can be used for joint ownership, but only if this is implied by the context.

Solution 2:

You could use our or his.

Their is now standard for a singular possessive where the sex of the referent is not important, but because you are speaking amongst yourselves about yourselves, use our in much the same way:

By tomorrow, one of us will see the money on our account.

Alternatively, use his as a gender-inclusive pronoun, or her if you both female:

By tomorrow, one of us will see the money on his account.