At the very beginning of the Minions movie, we are introduced to Norbert, coming out of the water wearing nothing, realizing his nudity, and covering himself... inappropriately. The voiceover states that he's a bit of an idiot. This scene clearly takes place thousands of years ago.
Later in the movie, we see a photograph set in Kevin's wallet that includes Norbert, with his name at the bottom of the photo, and text (or possibly speech - I forget which) saying he's an idiot (or some similar term - again, imperfect memory).
While it's only one character trait, it's the only Minion to have this particular trait applied. It seems to be intended as a repeated joke about the character, which only works if it's the same character. The character also has the exact same name. While it is possible that there are two Norberts, and both are idiots, this would break the intended joke, it seems much more likely that they are intended to be one and the same.
Similarly, other minions seem to exist both thousands of years ago and in 'modern' day. For example, Kevin appears to be looking over the edge of the cliff after the boulder... finished, and is back later in the 'modern' part of the movie.
This implies that Minions live a very long time. How long do Minions live - what is their life span?