This was actually possible and worked in older patches, but was broke in some 1.8 snapshot I'm not sure when

Steps to Reproduce:
1. Create a record with a custom name:

summon Item ~ ~1 ~ {Item:{id:"minecraft:record_13",Count:1b,tag:{display:{Name:"CUSTOM NAME"}},Damage:0s}}

2. Place in record player
3. Eject it, it no longer has its custom name

Checking the record player's NBT, it actually loses the custom name when placed in the player.