Ok, part of this idea was baked by the Babylonians, but there are a couple of modern additions that make it a neat idea.

First, convert base-60 to balanced notation (well, semi-balanced, since to be truly balanced it needs an odd base) -- instead of symbols corresponding to 0-59(decimal), you use
0-30(decimal) and a negative dot that can go under the number. To illustrate using the letters A-T as 10(decimal) through 30(decimal) and a minus sign instead of a dot, the numbers from 0-60(decimal) would be:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, 1(-S), 1(-R), 1(-Q), 1(-P), 1(-O), 1(-N), 1(-M), 1(-L), 1(-K), 1(-J), 1(-I), 1(-H), 1(-G), 1(-F), 1(-E), 1(-D), 1(-C), 1(-B), 1(-A), 1(-9), 1(-8), 1(-7), 1(-6), 1(-5), 1(-4), 1(-3), 1(-2), 1(-1), 10

This shrinks the multiplication tables normally needed for base-60. It also reduces the number of digits necessary to display repeating "decimals."

Next, we link each sexagesimal symbol to a phoneme (Thanks -- and apologies -- to juuitchan3 and his "Alphabet Base"). In English, there are around 36 pure phonemes, so if we set aside one for each symbol between 0-30 (decimal), we have 31, with five left over. We can represent the remaining phonemes as the following:

1. A negative dot underneath a symbol (by itself, subtraction, inverted it is addition)

2. A number dot that goes in front of a string of symbols (normal words do not have dot)

3. An inversion dot over the first symbol, which either means a proper name or (in a number) an inverse operation.

4. A symbol that stands for multiplication (inverted, division).

5. A symbol that stands for a power (inverted, a root)

The result is a combination phonemic alphabet and number system. Phone numbers would be handy for businesses and easy to remember for normal people. A six-digit number would be 2-4 syllables (usually) and long enough (60^6) to give almost everyone on the planet 10 phone numbers. The most common numbers -- 2, 3, 4, 5, 6 -- would all go evenly into the base, simplifying many mathematical operations. All sorts of clever mnemonics would come out of treating words and numbers similarly, and spelling would be a lot easier.