So I've done some more research on the topic based on what is stated on the Overwatch FAQ:
How do investigators get selected?
Investigators are selected based on their CS:GO activity (competitive wins, account age, hours played, Skill Group, low report count, etc.) and, if applicable, prior Overwatch participation level and score (a function of their accuracy as an investigator). Community members who maintain both a high level of activity and high Overwatch scores will receive more cases to elect to participate in.
From my investigation, the common scenario for people getting selected to participate is 150 competitive wins, which I passed a few weeks ago before being added to Overwatch.
With regards to hours of game time, I had just over 500 hours when it was enabled, but I don't believe this is the boundary. There probably is a minimum number of hours, but I think it is less that 500 as people on this thread have stated:
...had neither 500 hours nor AK+ ranks. I was Nova 2 when I got overwatch. Ursus
If it helps I got mine directly after 150 MM wins, at master guardian elite rank and just shy of 400 hours played in steam. .Johan #silverkrigarn
Got overwatch , gold nova 3 , 374 hours and 162 wins. Captain C. Caliber
neffer_23 stated in his comment that it needs to be 300+ hours, which sounds plausible but I can't verify that.
I'm currently Gold Nova 2, so perhaps the skill group needs to be Gold Nova at least.
I do play regularly and honestly I'm not that great so I would assume that I have a very low report count. I very rarely do anything worthy of someone suspecting me of cheating.
Also I've never had any bans, I guess that would definitely be a black mark against your name.