Blurred Lines on screen, then computer shuts off 9 months ago, my Macbook Pro 13in (2012) had blurred lines on the screen and  would immediately shut down. I went to Apple Care and they quoted me $700 to fix the issue, which I think they said was the logic board.  Today, I reset the NVRAM/PRAM by pressing Command + Option + R + P and it's been an hour and the computer is working just fine. Is it likely that the logic board is the real problem and is it work investigating/replacing myself? I am about to gift this laptop to kid who needs it for school and I don't want to give him a broken laptop. Thank you in advance.
I went to Apple Care and they quoted me $700 to fix the issue, which I think they said was the logic board...Is it likely that the logic board is the real problem?


If apple ran in house diagnostics on the machine and told you that you had a faulty logic board, then the logic board needs to be replaced. The issue could specifically be the with the integrated graphics component of your CPU, but it's a moot point since it still requires that the logic board needs replacement. 

This is a hardware issue for which no troubleshooting can ameliorate the problem or make it disappear. This problem will only get worse; inevitably (and without warning) the logic board will completely die and the computer will not boot. 


  Is it worth investigating/replacing myself?


That's subjective. According to iFixit, the OEM part costs either $400 or $600 dollars depending on which processor you have. In addition, factor in the cost of necessary tools and difficulty of fixing it. Draw your decision based on the cost/benefit of doing it yourself vs. taking it to Apple. 

Best of luck.