I don’t think you can get a burn with that light. It has a stepdown feature. I would recommend it to anyone that I deem competent even if they have no experience with li-ion.
With what many women carry in their purses, I wouldn’t want a light in there which developed much heat, one which switches on without a very deliberate effort, or a twisty which the jumbling of contents could unscrew. Women like pretty things and they generally have more acute color-vision so a CRI of 85+ will make it far more appealing.
For a tiny light the stainless BLF348 AAA is perfect, all the ladies who got one from me at Christmas love theirs. Only a keychain ring (which it lacks) could improve it. Use a Lithium primary cell to avoid the dreaded “alka-leak” corrosion problems. I don’t have enough experience to recommend a larger AA light but use the same guidelines above and she will like it. Anything heavier will probably not stay in her purse for long.
LiIon rechargeable lights are a whole different game and I would not recommend one for casual purse carry unless she fully understands the care and feeding these absolutely require. LiIon’s are lights for Pro’s and Hobbyists, not casual or occasional light users.
Phil
I gifted some BLF348s to women. Even included eneloop batteries and a USB NiMH battery charger for each.
My wife has a hot S2+ in her purse next to her handgun. I’m sure she can handle it.
Quite heavy purse she has. 
Hi DaveK14,
I’m a knife guy too, check this out (http://chineseknives.proboards.com/) it’s moderated by a fellow BLF member.
For birthday/holiday gifts I purchased a handful of these AAA Bronte BT01( Introducing the Bronte BT01 - full review), sweet little well made light available in various colors.
My current/favorite edc is the new AA/14500 Manker E11 ([Review] Manker E11 XP-L AA/14500)… very compact and super bright. The only thing it needs imo is a lockout feature… maybe V2.
Astrolux A01 in several colors?
I de soldered 2 of the 6 7135 chips on the NW S2+ and now it won’t get hot on MED.
Gonna remove 1 more, swap the tube for a 18350 one and that will be the purse light of my wife (among her pens there is a BLF 348 with engraving but she wants more power)
The red Convoy S2+ has a 60° diffusing optics, magnet (that looks very nice with the metal button) and clip.
I Suggest checking out the brand new Klarus Mi7
Not exactly a budget light. 
Well, it depends…
What is budget ? under 20 ? under 30 ?
Who sell it to you ?
Do you agree MAP ? or look for alternative vendors ?
Might take a look at some MAD deals :innocent:
Until the OP answers a few questions ebrytghing that is suggested is just a guess as to what may be useful to him and his friends.
heres a brief summery with battery wise,
Best 18350 : Manker E14 (Same as Astrolux S41) be cautios its get really really HOT (USD 45/ With help of BLF members USD35 possible)
Best AAA : Astrolux A01 (USD9)
Best 18650 : Nitecore MH20 (USD79 / With help of BLF members USD45 possible)
Best Budget 18650: Nitenumen NE01
Best AA: Sorry I have no idea
My reccommendation Manker E14 with 18350 Beautiful/ Powerfull
Manker E14 (s41) is to heavy for my pocket. 
Best AA: Jaxman E3 with Nichia 219B
Best AAA: Astrolux A01
Best 18350: Convoy2+… Astrolux S1
Best 16340: On The Road M3 with U2-5C
here is
a thread about favorite AAA and AA lights under ten $ lots of choices .
Pick one you like the look of and them maybe ask a few members advice about it …or do another search in blf
Do a search for this post. ThruNite T1 HI for just $15.95. Normally $24.95
I’d give you a link but I’m still computer illiterate
Beautiful high quality light for a steal of a price
I prefer a AA-sized EDC light for my EDC. Compared to a AAA-sized light, you get more output and longer runtime, while still keeping the light small and lightweight enough to be unobtrusive.
How about a ThorFire TG06 or TG05?
https://www.amazon.com/ThorFire-TG06-Flashlight-Tactical-Included/dp/B00UCY79ZS/
Like SawMaster suggested earlier, for purse carry where it might see infrequent use, I would suggest the use of Lithium Primary (non-rechargeable) cells to prevent leakage or meltdown.
+1 on the TG06, nice solid light that muggles can run on tried and true AA’s but if so inclined they can up the performance by going lithium ion.
SMM
True. I think this applies to non-technie people in general. With my Astrolight A01, i’ve had people ask where the switch was and comment that the battery is low because they didn’t know there was multiple modes.
BLF348 is probably what I consider perfect. One mode, simple switch, lightweight.
If it were an AA with multimodes maybe the Thorfire TG06.
I have yet to meet a person that knows how to use a twistie. Everybody will either just look at the light until I say something or try to unscrew the head for on. Reverse clicky is like some kind of magic to most of the people. No matter how I put it (camera shutter and focusing analogies), my A6 ends up in turbo.
Fact is, most people don’t expect a light to have modes…So a reverse clicky will be powered in whatever mode comes first.