i would recommend the Princeton Tec Attitude, but i don't know what its availability is like where you're at. except for that, it meets most of your stated requirements.
The L2R does sound good, and then he can choose from the wonderful array of P60 dropins. Another option would be the Tank007 TK-567 with the 2xAA extender.
Well, L2r needs foil wrapping but maybe he can live with that.
An advantage towards L2r is, that he needs also a camping light. Solarforce has that camping lantern head available.
At the moment he says he has a multi-LED 3xAA camping light, which sucks up batts too fast. Coleman -style camping lantern (D-batts) sounds to be too big.
Romisen would propably be fine but not sure about that strobe, might suggest that one.
e: added "relatively" to topic. I think it doesn´t need to be the most budget choice of them all.
Shiningbeam's Romisen RC-N3 II has been the 2xaa budget light staple for a long time. Give it a look. Its two modes with no strobe. You can also now get Terralux Lightstar 220's for under $25 so that might be another option.
If you run a L2R with just an XP-G or XR-E drop-in, you likely won't need to do the foil thing. With only 2.4 volts running from 2 nimhs, heat is not that much of an issue. I'd say get the L2R with the 3-mode low voltage XP-G module.
if you want to really cheap out, there is always the venerable, but flaky, C3.
The C3 is a nice size with a single cell, but I'm underwhelmed with its performance with alkalines. I like it in the single-cell with a 14500 which the OP wants to avoid. To me, the extension for the second cell really messes up the looks of the light, and makes it too big for the amount of light you get.
It's not a multi-cell, and it doesn't have modes, but the Romisen RC-G2 is the one I like to gift for people. It's reliable with a good switch, and really amazes the non-LED users when you compare it against a full-size Mag or similar.
I would agree with keltex78, I personally don't like the look of 2xAA lights. Too long and unwieldy. Unless a new user wants 2xAA for some reason, I would recommend 1xAA, my personal favorite.
Possibly the 2xAA is not very strict limitation and 1xAA might do just fine also.
I will recommend these as options, Romisen RC-G2 is also a good example what didn´t come to my mind even though it appears almost in every cheap-good-1xAA thread...
The RC-G2 is my go-to light for gifting because it has a very good switch, nice brightness, good throw, and is reliable. I don't want to gift lights that will need constant attention to work.
if your friend is going to only use alkies, you should avoid 1xAA lights. the current draws are usually too high for a single alkaline to handle. you want the load to be spread over multiple cells for maximum operating efficiency.