I realized that some Boeing manuals recommend that the engine number 2 must be started first, is there any special reason for this?
The isolation valve in
Auto ensures that the Auxiliary Power Unit (APU) bleed air reaches the engine number 2 starter. Engine number 2 start Exhaust Gas Temperatures (EGT) are generally higher.
Engine number 2 is started first because it's on the opposite side of the air bridge. When starting with external air or power it is normal to start Engine number 1 because it is further away from the carts.
According to Flight Crew Operations Manual (FCOM) 3 Standard Operating Procedures (SOP):
Engine 2 is normally started first. It powers the yellow hydraulic system, which pressurizes the parking brake
Looking at the electrical schematic that Boeing provides, we see that Transformer-Rectifier (TR) 3 is powered from the R/H AC Bus. TR 3 is an important TR from which significant avionics are powered. Ergo, you would want to have this TR powered up and running first in case you lose AC power to the aircraft during start.
Because most people are right handed.
Let me expand a bit on that:
Our story begins in ancient times, long before aviation took off.
Early boats were steered with a steering oar attached to the side of the boat. You can see one in the following tapestry:
(image source: Wikimedia)
Since most people are right handed, the steering oar was attached to the right side of the boat. The Old English word steorbord comes from this steering oar, from which the term starboard for the right side of a ship originates. When docking at a wharf, the steering oar would be in the way. That is why boats always docked from the left side, hence called the port side.
Modern ships use rudders on the centreline of the ship, so they can dock from both sides. The terms port and starboard are however still used today.
What does all of this have to do with aviation? Enter the flying boat:
(image source: Wikimedia)
In the beginning of aviation, most passenger aircraft were flying boats because very few airports existed at the time, particularly ones with long enough runways for larger and larger aircraft. Flying boats could just land on a river, a large enough lake or the sea. This is also the reason for why we are using many nautical terms in aviation (like e.g. nautical miles for distances) and why pilot uniforms look like Navy uniforms rather than Air Force uniforms (see also What's the reason for pilots wearing uniforms?).
How did these flying boats dock when they arrived at a port? Of course with the port side towards the port. That is why passengers had to board the aircraft from the left side. This tradition lives on until today (see also Why do passenger embark on the left side of an aircraft?).
Before Auxiliary Power Units (APUs) existed, you had start a main engine to get electrical power (without draining the battery) and air conditioning. Since the passengers were boarding on the left side, you had to start a right engine first. Once all passengers were on board and the doors were closed, the left engines were started as well.
Today, most (maybe all) passenger airliners have an APU to power the aircraft (and start the engines). They can also be powered by other means at the gate (see What powers an aircraft prior to the APU being switched on?). This means one could start either engine first today. Yet, it is still common to start the right engine first, e.g. the Boeing 737 FCOM, Normal Procedures, Engine Start Procedure says:
Normal starting sequence is 2, 1.
So why do we still care? In part, due to tradition. But there are some reasons to start the right engine first:
APU: The APU uses fuel and this cannot come from the center fuel tank because that could be empty (see Why are fuel tanks in the wings filled first, and why are they used last?). So the designers have to choose either the left or right wing tank for the APU fuel supply. Since it was tradition to start the right engine first, they choose the left wing tank for this. So the APU starts to use some fuel from the left and then when the right engine is started first, it will help to partially balance the fuel.
Electrical Power: DSarkar's answer claims that starting the right engine provides power to Transformer-Rectifier 3, which will power the AC systems. This is however not really important since the APU will power all AC systems during engine start anyway. The 737 does not even need any AC power when starting:
In the GRD position, the engine start switch uses DC power from the battery bus to close the engine bleed air valve and open the start valve to allow pressure to rotate the starter.
Hydraulic Pressure: DSarkar's answer further says that starting the right engine is important for providing hydraulic pressure to the parking brake on the Airbus A320. However, all Boeing aircraft can fully pressurize all hydraulic systems electrically (and some pneumatically) before the engines are started.
In the end, it really does not matter any more today. Any Boeing could safely start either engine first. As a matter of fact, sometimes the pushback crew actually asks to start the inboard engine first when turning the aircraft during pushback because they will have to overcome less torque that way.
And there you have it: we start the right engine first because passengers board on the left because boats docked on the left because the steering oar was on the right because most people are right handed.
Bonus: Tail Mounted Engines
The tradition to start right engines first only exists for wing mounted engines. Aircraft with tail mounted engines typically start the engines from left to right. That is why the APU typically uses fuel from the right wing tank for these aircraft (e.g. MD-80, Boeing 717).