Replace logic board for 2012 13' Macbook Pro or buy new laptop?

"It will freeze and have horizontal black bars across the screen, and when I try to restart it, I receive 3 error beeps repeatedly while there is a blank black screen."

"To replace the hard drive (with a solid slate hard drive which would be an upgrade) and the ram, it would be $300"

The 3 beeps almost always point to a RAM problem.

Have you tried:

Reset NVRAM

http://support.apple.com/en-us/HT204063

I had a 2010 that wouldn't start w/ 3 beeps and a NVRAM reset would get it to start. My daughter was using it on her desk and never shut it down just slept it.

If that is the problem make him prove it before you invest the money.

"freeze and have horizontal black bars across the screen"

That's a classic symptom of a GPU problem to me.

Here's a recent post about the Apple Flat Rate Program (Depot Repair):

Apple $310 flat rate repair plan? Have any of you actually taken Apple up on this deal to repair your MacBook Pro/Air? Is it worth it?

Take it to the Apple Store to get it analysed and get a quote for a repair. If it is costly enough to make buying a new Mac more viable then do that. Honestly, if it does come down to a logic board replacement, it will most likely be very expensive i.e. the majority of the original price of the computer. So if that is your only option, I would strongly consider just buying an updated Mac and selling the old one for parts.

So if the repair is very expensive, ask the Apple tech if his store participates in the "Depot Repair Program". Under this program they will repair any computer for a flat fee of $300. He might have to talk to his supervisor because often most techs don't know about the program.

This is your decision to make. Not ours. We can't tell you what to do. You've also provided no numbers, criterion, etc. Even were we able to help you've provided no useful information to use.

It covers everything apart from accidental damage. There is no reason why you shouldn't be eligible for this, especially since you've already paid full whack for a faulty logic board previously.