Every now and then my cat Bellatrix (Belly for short) will bring my husband and I her green mouse toy while we are sleeping. So we will wake up with the toy on the bed. We assume this is her favorite toy as she always plays with this one and will want to play fetch with this toy. We don't have a problem with her doing this and she doesn't wake us up when she does this but it is just curious as to why she does this. I honestly think it's cute to wake up with her favorite toy on the bed. I do know that some cats think their owners are just terrible hunters and will give them "prey". But I'm wondering if she does this because she thinks we are bad hunters or if in the middle of the night she just wants to play but then sees that we are asleep so she leaves us alone or if she just wants to give us a gift. This is the only toy she brings us. Belly is also an indoor cat. I'm just wondering of the different possibilities she could be doing this. Does anyone else's cat(s) do this?
While I think StephenS' answer is likely correct (i.e. bringing a toy to ask you for plays), it's also possible that the toy is left there because it's where the cat sleeps and it kept the toy near its spot.
Our cats cuddle their favorite toys and softly play with them as they lie in a comfortable spot. One of our cats specifically hides her favorite toys in her favorite bed to keep them away from the other two.
I do know that some cats think their owners are just terrible hunters and will give them "prey".
It's not a matter of thinking you're a bad hunter, but more a matter of contributing to the household. You give food to them all the time (which they can only assume you hunted), so when they have caught something and aren't that hungry, they share their spoils with you as well.
Cats aren't as cooperative with researchers as dogs are, so we don't understand nearly as much about how they think. However, it's fairly obvious cats know the difference between toys and real prey (and hunting or fighting vs play), so she wouldn't be giving you a toy because you're bad hunters.
Most likely, she is bringing her favorite toy to you because she's bored and wants you to play with her; if she plays fetch with it when you're awake, that is almost certainly the case. My cats regularly bring me their toys when I'm on the couch, though so far not be in bed.
It's possible she's bringing you her toy as a gift, but such empathic behavior is usually only seen from cats when you're behaving oddly during the day, e.g. because you're sick or injured or upset, not when sleeping as usual at night.