Had a hamster when I was young - very cute, but not much fun as he slept all day and kept me awake all night!
DD had two gerbils when she was 8, both girls, and we loved them. They lived just over 2 years. They DO have tails, but really, they're not rat-like - very soft fur on the tails and they're so sweet, they may cure you of any antipathy to mice/rats (personally I have no problem as long as they're pets and not running wild). They come in different colours, black, white, golden, mixed - so you could find ones that aren't mouse/rat coloured, which may make them more appealing.
Gerbils are VERY easy to keep, like being handled, someone I know even managed to teach his gerbil tricks. They're curious and friendly, will always pop out to see what's going on if they hear you. They like gnawing, so don't bother with expensive 'gerbil houses' as they'll eat them. Give them bits of carrot and they'll get through volumes of cardboard - cereal packets, loo rolls (they'll enjoy using loo paper to make their beds). We used to get our gerbils to nibble a bit out of our Christmas cards as a 'signature' before sending them out. (yes, very sad )
We used to clean out the gerbils about once a fortnight. They're very clean and don't smell (warning, you may wind up doing the cleaning!)
Go Gerbils !
Finally, if you have a compost heap, you can tip the cage contents (minus gerbils ) into your compost bin when you clean them out. It makes excellent compost!