I have a few mattresses lying around in the shed. I think we might have three spare. Maybe one or two big thick rugs as well. Can these be used to soundproof? Even just a little bit?

It should help, cardboard and foam is also a cheap alternative if you're willing to go further.
Yeah I've see it done with egg cartons and rugs. Foam would be good too.
good luck
MIM Fender Telecaster
SX GG1 Junior (Les Paul Special clone)
Dillion Canada DXC 58 (DC Les Paul Junior)
Fender Joe Strummer Telecaster
Squier CV Duo Sonic
Epiphone Les Paul Junior Special
Squier Jazz Bass
Vox DA 15
Fender FM 212R
Something else came to my mind, I replaced the wooden door in my jam room with a heavy duty metal-ish door a couple of years back and it made a HUGE difference. Just a suggestion if you have some spare cash.