He needs to get a partner visa. My husband (then boyfriend) non-EU was here under a youth mobility visa (he is Aussie) then when his visa expired, he reapplied for an EEA unmarried partner visa.
I had to apply for a residence card at the same time as a Qualified person (even though I didn't need one at the time) but made the process easier for him.
We had to demonstrate we were an stable couple, so we sent letters from our family members, close friends, conversations online while he was away, photos, boarding passes from holidays we took together, bills, bank accounts (we had a joint account), etc; it was 3.5 kgs of paperwork.
Once you get to the point of sending everything, check with your council if they give the service of certifying a copy of your passport because once you both send them, you will be left without it for 6 months and unable to travel.
Also your partner won't be able to work during that time.