Why not the other body parts like legs? What are the reasons for preferring arms?
Blood pressure is often checked in lower limbs also. BP needs to be checked in all limbs if one is suspecting obstruction in the arteries. Takayasu's arteritis and atherosclerosis are 2 conditions that can result in unequal blood pressure in different limbs due to obstruction of arteries.
In ankle-brachial index test, blood pressure is checked in the arm and at ankle to detect peripheral arterial disease (PAD: http://www.mayoclinic.org/tests-procedures/ankle-brachial-index/basics/definition/prc-20014625 )
Also, if coarctation of aorta (congenital narrowing of great artery in thorax) is supected, pressure in lower limbs will be lower than that in upper limbs.
Aortic regurgitation (incompetence of aortic valve) leads to higher pressure in lower limb than in upper limb due to pressure wave reflection. Here also it is recommended that pressure is recorded in both upper and lower limb. The degree of change in lower limb correlates with severity of regurgitation (Hill's sign).
Initially, blood pressure should be recorded in both upper limbs. If BP difference in 2 arms is more than 15 mm Hg (systolic), tests (e.g. Doppler ultrasound) should be done to rule out obstruction. For following up on treatment of high blood pressure in such cases, higher blood pressure should be taken and controlled with medication.
Blood pressure can also be estimated using finger probes, but their accuracy and reliability is not well established. This will also be subject to diseases of arteries in the arm, forearm and hand.
For checking blood pressure, one needs an area where the artery can be compressed and one needs to listen to the artery distal to this area as the pressure is released. Brachial artery lies in front part of elbow and can be heard over easily. On the other hand, the artery in knee lies posteriorly (behind the knee).
Since arteries to upper limbs come out of aorta much earlier than arteries to lower limbs, the pressure in upper limbs would be much less subject to diseases of aorta or compression on it by masses in thorax or abdomen. This would be main reason to prefer arms over legs, in addition to simple convenience factor.
The reason we use the upper arm is that the arteries at that level have the same pressure of blood as the outflow-tract from the left ventricle (i.e. where the blood leaves your heart).
Because of the weight of the blood, your pressure in the main arteries in your leg is slightly higher and in your head is slightly lower. This effect is more pronounced when sitting upright or standing.
I think that we also measure it there for practicality - for serial measurements to make sense we should measure it in the same place every time, and an arm is much more easy to get to than a thigh (you'd have to take your trousers off every time!).
Measuring pressures at the end of limbs is only useful in very specific conditions (such as peripheral vascular disease).