I went to a new car dealership and signed all the papers to lease a
new Lexus and I was then told that the head rests needed to be
changed, and I would have to wait to pick up the car the next day, but
they asked me to pay for the 1st month of the lease which I did. Now
the next day, they said that they are short of mechanics and I may not
be able to get the car until tomorrow. Can I cancel this agreement and
get my $419.00 back? Or what?

Probably not. Tell the Researchers what state you are in and they may
be able to answer.

If there was fraud or a major loss from their actions, you may have a
case.  But I think a judge would probably laugh at you if you are
trying to recoup damages because you were delayed by a day in picking
up your new car.  If this delay has forced you to rent a car in the
meantime, then you should contact the dealer's manager and tell them
that you expect them to pay for the rental costs due to their delay. 
I think your maximum loss from this 1 day delay is probably around
$60, not enough to get you out of a contract.

If the dealer has not "delivered" the car to you because of defects in
the car, you quite likely are able to void the contract as not
fulfilled. The lease contract is essentially an offer by you to pay
money for use of the car. If you've NEVER had use of the car, there
has been no fulfillment.

This argument is far weaker if you have taken possession of the car
for even 5 minutes, but if you haven't no lender in the country is
going to accept the contract from the dealer.

If this argument fits your facts, skip the dealer, call the lender and
explain the situation to them. Tell them that you have rescinded the
contract, that it is null and void and that you'll not be making any
payments on it. Do so very politely, though, so they can carry your
water for you at the dealership.

You should also make the same argument to the most senior manager you
can reach, at least Sales Manager, better, General Manager.

But, again, this is dependent upon the fact that you never took possession.