For me personally, I thought Hamilton proved his skill when he went head to head against Alonso in his Rookie Year. Also, bagging all those podiums in his first races with McLaren (no matter how dominant McLaren may have been at the time) got my attention. Here's a rookie, bringing the fight to the great Fernando Alonso. Ever since then, I was convinced that given the right machinery, this driver was going to dominate in the same way Michael Schumacher did.
While he was in a dominant car in the Mercedes, I think it is also important to note his accomplishments outside dominating cars. I do not count the McLaren of 2008 as a dominant car because Ferrari got the WCC that year.
Also, I think Rosberg was an underrated driver. He had an excellent record prior to driving against Lewis. He is at least much much better than Bottas IMO. The outlier I think was where Button outperformed Lewis, but that can be due to distractions in Lewis' personal life too.
One last thing, I think Lewis improves after every major challenge. He's a much different driver to what we saw in 2016 and in my opinion he just keeps improving. I do not think he still possesses the single lap pace he did when he was younger, but I do think he is a more complete racing driver now.