Diagnosis
That's the drive shaft socket, and it's a magnet for toothpaste slurry and saliva, which hardens into a cement-like gunk.
How to Fix It
Unplug the base. You'll need a toothpick, a cotton swab, and some isopropyl alcohol. 1) Gently scrape around the inside rim of the hole with the toothpick to loosen big chunks. 2) Dip the cotton swab in alcohol and swab out the socket. Don't pour alcohol directly into it. 3) Use a dry swab to mop up the dissolved residue. 4. Let it air dry completely before attaching a brush head.
If That Didn't Work
For really stubborn, hard deposits, try the vinegar soak method. Dampen a small piece of paper towel with vinegar, twist it into a tiny point, and gently stuff it into the socket for 10 minutes to soften the minerals before scraping.
Heads Up
Make it a weekly habit. Every time you change your brush head (which should be every 3 months), take 30 seconds to swab out that socket. Prevention is a whole lot easier than cleaning a year's worth of buildup.