Since a couple of years, all Ubuntu flavours (including Ubuntu-MATE) have switched from MBR to GPT partitioning for both the installation medium and the target disk (in your computer).
In have experienced this "sometimes boots and sometimes not" situation with GPT on buggy non-UEFI BIOSes (most firmware is buggy) on computers older than 2012.
Also had GPT partitioned Live-USB sticks that won't boot at all on some computers.
I don't say that this is the cause of your laptops-shenenigans but it is one of the likely possibilities. (the other reason might be that either yout harddisk, your RAM or your motherboard is dying)
Here is some extra info:
In short:
BIOS bugs can potentially make boot from GPT pretty unreliable or even impossible.
Although you computer is 64-bit, you might choose for a 32-bit distro precisely for this reason.
(32 bit is never GPT/UEFI). Some good choices here:
For two of those laptops that had problems with GPT (including a Sony Vaio from 2006) I installed Bodhi Linux (the 32-bit version because the 64-bit version was all GPT and wouldn't boot at all)
The other good thing with Bodhi linux is that it consumes only about 130MB RAM so for a computer with 1GB Bodhi would be quite a better fit than Ubuntu-MATE.