Don't bother going to Asakusa to look for overpriced, unimaginative souvenirs. It's crowded and over-hyped!! If you're feeling adventurous, walk along the small lanes and streets and look for mom & pop stores that may have something more interesting.
Unique Suggestions: Just look around in Asakusa and take lots of pictures!!
Fun Alternatives: Look no where else other than Don Quixote discount store in Kabukicho (5 minutes from the JR Shinjuku East Exit). There's also another one on Shokuan Dori, behind Kabukicho, near Shin-Okubo station.
You could spend hours and hours in there looking for foods, snacks, clothes that real Japanese eat and use. Have fun!!!
We are TOURIST and being TRAPPED inside this station with endless tunnels and exits for almost an hour, knowing nowhere to go. Even local peoples seemed get lost too!