Hard to see. Is the battery female threaded (recessed threads) and the cartomizer is male threaded (a threaded metal part sticks out)? And it is s 2-piec ecig? Just battery and a cartomizer (a
juice cart with internal atomizer that is one piece - it holds
juice AND screws right onto the battery?) BTW grats on manual batteries if that is what you have. The are much harder to kill and make it easier to get a good hit.
If at least 9.2mm diameter then it is probably RN4081 threading or 510 threading. But it could also be the mystery female threading nobody has figure out (some incompatible clone). You can get some RN4081 cartomizers to try at some 7-11's (called Njoy Npro or Njoy 2-n-1) or online at Njoy or Smokestik or AZSmokeFree. if they don't fit, then try a pack of prefilled Boge 510 cartomizers from Vaporkings.com (Chinese juice, there are few prefilled cartos for 510 but lots of good blanks, Boge being the most popular and slim ecigs must use the 3 ohm Boges). If neither fit then it may be the mystery model.
And warning - if you figure out what threading it uses (i.e. which cartomizers fit) that does not mean that you can charge that model's batteries in your charger or vice versa because some clones are reverse electrical polarity and can fry normal batteries in their charger and vice versa.
And note - although trust called what you have cartridges, I think they are cartomizers. Cartridges aka carts are used on a separate atomizer piece. Cartomizers are 1-piece disposable combined cart and atomizer and their screw right on the battery. If you can carefully pop out the mouth end cap and see stuffing, you can add
eliquid to the stuffing to keep a cartomizer damp. I can get up to 10 days use out of a good cartomizer using thin juices (PG with 20% or less VG). but some "name" brand cartomizers are cheaper ones that don't work for many refills. if you let a carto get too dry and singe it, it will get a bitter taste.
!!!!!!!!!!! **IMPORTANT*** !!!!!!!!!!!!
AND IMPORTANT!!! Make sure your mom didn't sign up for a subscription where she will pay every month or week for cartomizers!!! Some people have been ripped off for thousands of dollars by subscription deals. They lie about how long one cartomizer lasts, don't supply enough for the subscription or charge more than they claimed, and if you need more cartomizers they really rip you off.
If she signed up for a subscription she should cancel it!!!!
There are fine ecigs that are not a ripoff. Like Liberty-Flights.com is a great multibrand vendor. Press the 20% off upper left front page then shop. They have Joye (real) 510 manual kits and they have blank Boge cartomizers (3 ohm is for the slim 510) and with manuals you can use a drip tip on the atomizer to test juices. If you want it to look like a cig, get a white manual 510 and the tan or white Boge cartomizers from NiteLiteVapor.us. Liberty-Flights.com has some juices and there are dozens of good juices vendors on the web. (for example, FreedomSmokeUSA.net or for cheap Chinese juice Madvapes.com has great prices)
Or instead, look at KR808D-1 at Vapor4Life.com or another style at SmokelessImage.com - this is a good model for using prefilled cartomizers (you can keep adding juice to them to get a week or more use). Again I recommend manuals.