Alright powering the Arduino board with 9V (through the jack) is causing the reset. Does this mean that the regulator is failing? because for what I've readed powering with 9V through the jack is even advisable.
Is it a bad idea to think about powering with 5V through the jack? because that's how it spent the night and it was all steady. EDIT. It is a bad idea I just did it and the ethernet shield does not start. What about trying through the USB connector?
Yes. 6.5V minimum for the barrel socket. With 5V supply, use the 5V pin.
Hard to say what you did wrong there without diagrams and/or pictures. But it is perfectly normal and to correct to power the mega with a 5V supply connected to its 5V pin.
Thanks. You gave me trust to try it again and it worked fine through the 5V pin. Then probably I used the 9V adapter (instead of the 5V one!!) in the previous try 
their terminals look almost identical.
I will test this way for some hours, then using the relay shield and I will update.
EDIT. I experienced resets with the shield and using the 5V pin too. I'm back to power through the jack using only 5V, I know is not recommended and that the ethernet shield wont even start but is what has worked for now (like no reset during hours) so I gotta start from there to see what's wrong.
If I don't experience resets I will try again powering through the 5V pin.
One of a number of really stupid "features" of the new forum system. It may have its uses, but these "features" should automatically turn off and disappear after say, 100 posts because this is a technical forum, not a social media site. 
I get you. I'm not a regular user but I remember the old forum and it was more intuitive. I may need to explore and get used to this one tho. I don't know how to answer at a specific post like right in its below, I saw it and I liked it, but everytime I answer it goes straight as the last comment. Wait I just noted some of my answers are right below of the comment, like I wanted, I'm guessing is just a matter of clicking the reply button of the specific comment and I guess one couldn't answer to several comments in the same post because the system would send it to the bottom ¿?
NOTE. I searched for a tutorial for the new forum and could not find it.
Hi.
I'm having issues with powering through the 5V pin.
My "safe zone" is powering the Arduino board through its DC jack using only 5V, this way the board won't reset and will be steady. I cannot use the ethernet shield because of the low voltage and neither am using the relay module. This way has shown to be steady in time, no resets. Then when I increase the voltage to 9V it starts reseting and if I power with 5V through the 5V pin happens the same after a short while. I haven't tried the USB port tho. I wonder if there could be something wrong with the board or one of its components.
I wanna try powering the Arduino board alone with 5V through its DC jack and all other components through the LM7805 but I just noted it would be difficult with the ethernet shield due to its fixed pins
It's not really "safe" at all. The Mega is probably running on as little as 3.5V, and the chip is not guaranteed to run at 16MHz on such a low voltage. This won't cause any damage, but could lead to random freezes and unexpected crazy behaviours.
I also wonder if your components are faulty. I can't think of any other reasons for the way your circuit is behaving. I suggest taking everything apart and then find ways to test components in isolation, one at a time.
I tried powering through the USB port and it resets too.
Thanks for the advice. I've unplugged all the parts and started from cero using the millis() example. It's steady so far. I will perform a clean up over the board surface with an electronic cleaner. I once had issues with the USB / COM interface and solve them with a deep clean. I will start testing each part and will update with results.
I have some important update about this topic. I started from zero testing each part by separated (as adviced by @PaulRB, thanks!!) and found that the cause of the resets was the LCD. I'm using a steady version of the code, so I thought about quickly modifying it to replace the LCD sentences for Serial.print ... and it works really well. I will replace the LCD for a new one and see how it goes.
Hi,
Can you post the version of your code that has the reset problem?
When you omit the LCD software in your code, do you still have the LCD connected to power?
A picture of your project would also help?
Thanks.. Tom... 
Oh I'm so ashamed. Please forgive me on this one @PaulRB .
I've listened to @TomGeorge when he said:
I ran the program using both USB/COM and the LCD as interface. With the LCD the reset started to happen. Then I just started testing with minimal LCD pins connected... and thought about connecting the potenciometer since I had one already connected in the board. After a while the reset did not happen! and that was yesterday. Again I'm very sorry was this connection (LCD pin3) to GND causing some sort or short circuit? does the pot actually never delivers 0v?
I have another question, please tell me if I should open a new topic or maybe just searching within the powering existing topics...
When I tried the USB/COM and LCD simultaneusly I noticed that the LCD was brighter than ever and I really liked it. But then I realized that the board was being powered simultaneusly through the 5v pin. Is this a bad connection? why powering only through the 5v pin affects the brightness of the backlight and the contrast? I also noticed that using the relays decreases the backlight a little bit too. I'm powering the 5v pin with a AC/DC 5v 1A adapter and the breadboard with all other components with a 9v 1A through a LM7805 regulator.
Hi. Thanks!! uhm I don't think its a software issue, the code has +1000 lines!. I gotta wait a few more days but I think the reset problem is gone now... thanks a lot for posting about this topic, it seems that another user had already pointed the cause of the issue, I should had listened better!!
There is a Fritzing one at the beggining of the topic or you mean a real one?
Hi,
Yes a real world picture(s) would be great.
Tom....
Usually it is better to continue with your existing topic. If you start a new topic, other forum members will only ask the same questions and ask for all the same details of the project over again in order to be able to help you.
Having no potentiometer would make the display difficult to read, because the contrast cannot be adjusted, but it would not cause resets. Adding the pot would not prevent resets. So I do not think adding the pot fixed the resetting problem, even though that is how it may appear.
I cannot picture in my mind what you are connecting there. Please post a schematic of that.
And will report it as a "cross posting" so that a moderator can connect it to your original post to keep all the critical information together. You will get a warning about wasting people's time. 