Quoted:
The answer is "it depends".

Is your buddy doing the engraving "for free" for his "friend" as a favor than no he does not need an FFL.  

If this is a friend doing a friend a favor (i.e. gift) situation,  and depending upon your State laws you don't even need to stay with the firearm (assuming it still in a Title 1 configuration) as friends can loan, borrow, etc. firearms to other friends all the time to shoot, hunt, admire, or if they want do a favor and engrave it and then return it.

Now, if your buddy is charging you for fee this engraving service than he may be considered doing it "as a business" and doing any gunsmith service "as a business" per the BATFE requires a license.  It doesn't matter if the owner stays with the firearm or not.

https://www.atf.gov/firearms/qa/license-needed-engage-business-engraving-customizing-refinishing-or-repairing-firearms

You could probably argue there is some gray area between doing it for free as a friend for a friend, and charging a friend for a one off service, and doing firearm engraving as "a business" .  Similar to the situation where you sell a firearm you no longer want to your friend vs. gifting your friend the firearm and you are not "in business" even if you sell it, but where that exact line of where money starts to change hands for a one off deal for a friend ends and where "in business" begins isn't exactly 100% legally defined.

That said if your buddy charges you $25 each to engrave a handful of your guns or you bring him a $100 bill or a nice bottle of whiskey and neither you nor the friend plan to go rat each other out to the BATFE, the odds of anybody getting in trouble or prosecuted by the DOJ in this minor firearm engraving transaction are statistically about close to zero.

Prior to the proliferation of "lower" cost laser engravers, I used to do all sort of NFA engraving for AR15.com member in central Texas using my personal rotary pantograph machine just as a community service and never had anybody pay me for the engraving service and I would do it as they waited.

I no longer do the AR15.com engraving for free service as there are tons of local services now available who can do it.  However, I still do all sort of gunsmith type stuff for local friends.  That said, I never charge them any monetary fees for the work as I don't want any impression that I an "in business" of gunsmithing.   That said sometimes they buy me lunch or bring me a nice bottle of booze for my birthday down the road but there is never any obligation for payment.  I gift them a threaded barrel and the next time we go shoot they may buy me a burger and a beer afterwards, etc.