Well, if not your strap, you could put your weed in there.
I've heard accessory cutout spaces referred to as drug compartments, but didn't know anyone actually used them for that purpose until I joined a band full of stereotypical Deadheads. On the night I auditioned, the guitarist (complete with a replica of one of Jerry Garcia's guitars), pulled a baggie out of his. I made a joke about that being the drug compartment, and felt a sense of dread when he laughed and said that's exactly what he used it for. Everything went downhill from there as they all got stoned and the songs turned into painfully long jams. Most ended with the guitarist saying "That was sooo cool, maaan!" I tried to give the band a chance, but bailed after 3 weeks. I then joined a band with my old drummer and a drunk guitarist, and gave up on that after 5 months. Drummer and I are now in a band with a guitarist who's a drug and alcohol counselor. He puts picks in his drug compartment.
I have the same gripe as the OP because I use 3 and 4" straps. As much as I love the old Peavey chainsaw cases, I don't like the tiny drug compartment and lack of space for a strap. I toss my straps into a duffel bag that has my extra cables and gaff tape. The best cases for storing other stuff are the non-form fit one-size-fits-all, but I'm not crazy about the fact that most of my basses slide around a bit in them.
