As the local physics student, I feel compelled to chime in on this one and go all scientific on the discussion.
First, the solution somewhat depends on what you mean by a day. If a day is one day-night cycle, just living at a pole is enough to lengthen it considerably - the poles spend half the year in darkness and half the year in light due to the earth's axis being inclined. Now, admittedly, there're short transitional periods around the equinoxes so you can't get a full year, but a day-night cycle still lasts 6 months if you play your cards right. It's notable that running around the pole won't help you at all, since the entire top 10% (approximating off the top of my head, here) of the planet spend some part of the year in complete night or day. 'course, with a faster vehicle than just your legs and a position closer to the equator you could spend an indefinite amount of time in day or night.
However, this is a rather silly definition of day, since it makes the concept very subjective and quite prone to geographical variation. Typically, we just define a day is the time taken for one revolution of the earth, which is the same for everyone (well, everyone on earth, at least - but more on that later). You can therefore make the day longer by slowing down the earth's rotation. This is sadly not an easy thng to accomplish. The potential energy of the earth's rotation is given as
E_p = L^2 / ( 2 * m * r^2 )
where L is the Earth's angular momentum, given by
L = 2/5 * m * r^2 * w
W is the earth's angular velocity, which can be derived from the time of a single revolution as
w = 2 * pi / 85 823
85 823 is the number of seconds in a day.
This gives us a final expression for the potential energy in the rotation as
E_p = 8/25 * pi * m * r^2 / 7365587329 = 2,606361399*10^30 J
with good values for m & r
Now, assuming we're going to slow down the rotation from earth and not start slamming asteroids into the planet, something that would make living here uncomfortable, we're going to have to accomplish our task by discharging mass from the surface in the opposite direction to the rotation. For example, we could use the world's oceans as rocket propellant. So, let's say we make two large rockets on opposite sides of the globe, connect them to the nearest ocean and start blasting water out into space. Let's also say that we're shooting it away at the good, nippy and totally unrealistic speed of 3 000 m/s. If we had our rockets thus drain the world's oceans of all their water - 1,4 * 10^21 kg, all in all - and eject it into space, we would have spent 4,5 * 10^27 J slowing the earth down, according to the
E_k = 1/2 * m * v^2
formula for determining kinetic energy.
Assuming all this energy went were it should, instead of generating heat by friction or somesuch (and ignoring the effect of changing the mass of the earth by removing all the water), we would then have created a change of 5% in the angular velocity of the earth - not much to brag about, really. We'd need our water rockets to fire at about 30 000 m/s to halve the rotational speed, and 60 000 m/s to stop it.
There are other physically possible but more esoteric methods that can slow the rotation down. Special relativity has time dilation, which basically states that observed time slows down for things that have a high velocity - nearing the speed of light - compared to your own. You can also use an effect of general relativity that has time slow down when space-time is bent out of whack near strong gravitational sources, simply place the earth near a black hole or similar object and place yourself very far away from any such strong gravitational forces. Finally, quantum uncertainty allows you to be at multiple points in time with varying probability when at low energy levels, so freezing yourself to near absolute zero could concievably work as well. However, these methods are solutions that would only serve to make longer days for you and not the rest of the earth - indeed, one of them would probably destroy the planet - and thus I cannot recommend them.
Edit: D'oh. Code tags didn't really work as intended, but I'm too lazy to fix it :p
Also, I take no responsibility for embarassing calculatory errors. Those were all made by my evil twin.
