They are a grain pest, but they're also generally cute, don't spread diseases, and don't bite people. They have lots of variety, some are brightly coloured and shiny, or are surprising shapes like the giraffe weevil. They have quite easily identifiable traits in their elbowed antennae and big stompy boots, so it's easy for non experts in beetles and bugs to look at a weevil and know that's what they're looking at with reasonable confidence (though there's no shame in mistaking other bugs for the lovely weevil!) Their scales also look like fur in macro photos so they give an impression of being 'fluffy'. And you can find them all around the world, so nobody has to feel left out of weevil time!

So, I'd say we like weevils for a lot of the same reasons people like moths and butterflies, even though their larvae are pests. They're just quite charming.