Not far from the Main Marks & Spencer store on Oxford St is a place called Reubens its on Baker st #79 Fleshig. If I remember correctly Selfridges is across the street from that M&S and the street that is between is Baker St. Its a ways up not a long walk or hop a bus (I think Baker is a 1 way going the wrong way, so youd have to go 1 street over grab a bus over there and then walk back to Baker st. Or simply walk up Baker for appx 5-10 mins. If walking from OxfordSt it will be on your left side.
If I was you Id buy a day pass for Zones 1-6 (LHR is in Zone 6) that way you can travel on the Underground and the buses as well
Or simply take the Nothern Line to golders Green station or Hendon station and there will be more places to eat then you will be able to decide where to actually eat. GG will be closer, Hendon will be a bit of a walk, but I prefer the restaurants in Hendon. Or take the #13 bus to GG from Oxford st it ends at the GG underground station. It used to be the old Roadmaster double decker buses, Boy I miss them. Its a great ride and you'll see alot not much Touristy but I love the ride on the 13.
What ever you do, DONT stay at LHR unless its pouring rain!