Maximas are know for starter problems. You need to replace it. (You might be
able to fix the one you have if your handy with tools and it is fixable) The
EXACT same thing happened to my daughters 96 SE while she was staying with
her dad (I am her step-dad) and he did the same thing, first bought a new
battery,  NO-Joy, bought a starter and it did the trick.
HOWEVER DO NOT buy a "parts house" starter, get a factory one from Nissan,
yes it is expensive, but not as expensive as replacing it 2 to 5 times (he
went through 3 before he got one that works reliably....at least it is still
working now...(fingers crossed)) with a parts house starter, and you may go
through that many to find a good one, yes they will have a "lifetime"
warranty but you or your mechanic will be changing it, (labor!) not to
mention the down time.....
Good Luck
2Maximas
(96 SE (daughters), might as well be mine I bought it and fix it, unless I
get lucky and she is at her dads house like with the starter situation! and
97 SE)

Replace the relay first.  If you hear it click it just means that it is
being activated.  You have no way of telling if the contacts in the
relay are good unless you get a multi-meter and check the volts that
come out of it.  The car is almost ten years old and my first and
cheapest option is the relay.  After that you will be replacing the
starter.  Somethings you did not mention are if the battery checked good
and if the starter has ever been replaced.  Remember that the starter is
a gear reduction unit and as such they tend to dry out and the older
they get the more current they require to turn over.  If the brushes are
worn and the grease has dried out it would take a few attempts to get
the engine to turn over.

This has been a frequent problem that I have too.  And my 95 SE is over 10
years old (purchased May 94).  The symptom is you can hear a relay click
once when turning the key, but no crank.  But I wonder if there are
multiple contacts on the ignition switch, because if I jiggle it while I
turn it, that usually works.

I have been waiting to see if someone who had this problem figured out
what it was and resolved it.  But if they did say, I missed it.

I did replace the starter years ago.  At that time it seemed like it had a
bad winding (loping crank).  I replaced it when it apparently stopped on
the dead winding and would not crank at all.

Finally, Thanks all to help me. I replaced key switch and it is working
fine now.