Is there any reasoning that I could use to defend my answer to the
following question:

Question: A hydrogen atom has a proton and an electron. If the orbital
radius of the electron increases, the potential energy of the electron
________

My answer: Depends on the zero point of potential
Claimed correct answer: increases

Your answer is incorrect. The zero point of the potential never
matters for anything. You can set it to whatever you want, at your
convenience. For the Coulomb potential the convention is to set the
potential to zero when the electron is infinitely far away. Suppose
you choose instead to set the potential to Vo at infinity. Then the
potential at a radius r is given by:

V = Vo - e^2/r

The larger you make r, the larger the potential energy is, no matter
what value I choose for Vo.

Think about it. The electron is attracted to the proton. I have to use
energy to pull it further away. Thus the potential energy increases.

The first time I read your question, I was confused. I didn't realize
that the separator line was meant to be understood as a blank to be
filled in. Now I see that your answer is not an incorrect statement,
it just evades the obvious intent of the question. Yes, certainly the
potential energy depends on the zero point, but clearly the question
is asking what happens when the radius increases.

I need a good way to explain the reasoning for my answer. do you have
any examples i could use to prove the question wrong or make my answer
seem more reasonable (help me explain my reasoning to others)?

I can't help you explain your own reasoning. I hope you won't take
offense, but if I were grading this answer it would not receive any
credit because it is a trivial statement that evades the intent of the
question, which is asking what happens when the radius increases. You
might as well have filled the blank by saying: "... the potential
energy of the electron is a form of energy ..." . In the words of
Wolfgang Pauli, "It is not even wrong."

You must make the distinction between absolute potential energy and
relative potential energy.  Your question is a bit like a hiker on a
hill. What is his absolute potential energy?  Well now, that depends
on where you define the zero point to be.  For example if the base of
the hill is defined as zero and he is at 100 meters above it. Then his
potential energy, U is

U=mgh

Where:
m=mass, of hiker let us say 50kg
g=gravitational constant=9.8m/s^2
h=height of hiker 100m
then:

U= 50kg ( 9.8m/s^2) 100m =approximately 50kJ 

Let us say he descends to 50m.  Then his potential energy changes to:

U= 50kg (9.8m/s^2) 50m =approximately 25kJ. 

His relative potential energy decreased, but at all times he had a
positive absolute potential energy.  If you set the reference level
very much above him, he will have negative potential energy but he can
climb and descend in other words, gain and loose potential energy.

So it is with the attraction between an electron and proton.  Instead
of U=mgh(for the gravitational field on earth), We have

U= - k e^2/r

Note the negative sign
Where:
k=coulombs constant
e= the fundamental unit charge
r= the distance between the electron and proton

You can see from the equation as r increases U increases.  (I figure
you are okay with that.) Here, I think, is the real meat of what you
are looking for.  By convention for particles of opposite charge
(electrons and protons), U=0 is defined to be at infinity.  You can
see this by inspection of the above equation.

Now if we pick some arbitrary distance to be zero, let us call it x,
then the potential energy equation becomes:

U= (k e^2/x) ? (k e^2/r) = k e^2( 1/x - 1/r)

Now if you make r>x then the potential energy is positive.  But these
does not change the fact that as r increases the potential energy also
increase.

Is anyone still reading this?

Oh yes BTW the answer to your question I am sorry to say is no :(

Okay here's the thing. I know that potential energy calculations can
get confusing because you can select any point of reference.

But you should consider this :

Since positive and negative charges attract each other "naturally" so
a more favorable state (less potential energy) of the system would be
when the charges are close together. So the correct answer to the
question would be that the potential energy "increases". No matter
what reference you choose , if you apply the rules for potential
energy calculations correctly then on increasing the orbital radius
for the H atom in  question the potential energy will always increase.
Trust me its not a very high funda question .. its basic physics.

Thanks,
Ankur

You might keep in mind that the definition of the electric potential at a
point is the work done in bringing a unit positive test charge from infinity
(obviously zero potential at infinity) to that point. It is also immediately
obvious that negative work is done in bringing a negative charge from
infinity to proximity of a positive charge.

kottekoe is completely correct. "zero" potential is defined as infinitely far away.