There may be a subtle heat effect occurring that close to the MagSafe connector.
Is the connector (both in plug end and inside the Mac) free from debris and clean
enough so the spring-loaded connections are able to adapt to their appropriate
counterparts with no interference?
You may need to consider asking an Apple Store for replacement MagSafe if
this component is independently overheating and those contacts are clean.
While my thoughts tell me the discoloration is due to heat ~ perhaps also the
unit may be trying to charge from a deep cycle discharge & uses more power.
For symptomatic relief, if the unit is generating excess heat while charging but
not having been greatly discharged, you may see if troubleshooting by use of
the SMC Reset instructions (steps - prior to actual reset - within instructions)
could help. ~ Use all the "Before you Reset SMC" steps... there are Nine.
• How to reset the System Management Controller (SMC) on your Mac - Apple Support
..These can be seen as troubleshooting non-invasive maintenance. Can't hurt..
Plus [ NVRAM or PRAM ] reset may supplement SMC; while not interchangeable.
The discoloration of insulation layer is symptomatic; heat likely may be the cause.
(note section on cleaning magsafe contact points among detail here)
• Apple Portables: Troubleshooting MagSafe adapters - Apple Support
• Reduce cable strain on your MagSafe power adapter - Apple Support
• Using and maintaining your Apple MagSafe Adapter - Apple Support