What is the cheapest way of getting a taxi from Barcelona airport?

The cheapest way to take a taxi, is not from the airport (which include the fees someone already mentioned).

Happen that just 1 week ago I had a friend who came at midnight too, so I know pretty much what would be the safer/cheapest way.

Your best shot is to take the Aerobus, that goes from the Airport T1/T2 to Catalunya square (center of the city) and cost 5.90€ (way chepear than a cab).

Then you can take a taxi (there are many around) and avoid airport fees (will be still lots cheaper).

I higly recommend you to do this, since it's very safe to take the bus (the only problem could be your arriving time, but if you hurry you may still get it

Ps: Something to keep in mind on Aerobus.

Ps2: If you think the bus is not an option, then you are at your own with taxis. Except if you want to sleep in the airport until 5 am (when the metro opens) or if you come on saturday when the metrol is open all night.

It really depends on the time you are arriving.

As others have already mentioned, you can use Aerobús to get to the city center. The last one is at 1.05, as they say in Timetable:

From 05:35 h to 07:20 h every 10 min
From 07:30 h to 22:20 h every 5 min
From 22:25 h to 01:05 h every 10 min

You mention

since the night transport is not too convenient

Well, it really depends on what convenient means to you: Aerobús is quite fast and goes straight. And all the options are safe: I have never heard about any safety problem in any of them.

Then, you can use a Night bus all night long. It is important to mention that it takes more time than Aerobús: around 40' instead of 20'. It is the number N17 and its current depart times from the Terminal T1 are:

21:55 
22:05 22:15 22:25 22:45 
23:05 23:25 23:45 
00:05 00:25 00:45
01:05 01:25 01:45
02:05 02:25 02:45
03:05 03:25 03:45 
04:05 04:25 04:45

If you really have to take a taxi, you'll find plenty of them and its fares are around 30€ at night to go to the city center.

The best way to not get scammed in pretty much every country in the world is to get a taxi with a meter, and/or ask how much it will be before you get on, so you know what to expect. The driver will know to give you at least an approximate value he should stick to.

Check online for the taxi fare ahead of time, but also consider that night-ride fair is usually more expensive.