Tell us how you do it currently and then will try to help.
Are you putting a knot in the cotton at both ends?
Tbh, I might just accept the MB's offer because it's much easier to learn when you can see it done in person and she might want it done in a particular way. Not everyone knows how to sew and there's no shame in that.
1. place name tape on item of clothing where you want it to be eg usually neck area or back of trousers and pin securely along middle of label with pin going the long length of the label not across it
2. thread your needle using white cotton thread or a thread that matches the item of clothing (your stitching won't show as much). Use a length of thread about 10 to 12 inches (25 cm) long. Tie a double knot in the end of the thread.
3. start sewing along one of the long lengths of the label by putting your needle through back of label in the middle of one of the long sides only going in about 1 mm at edge. Pull the thread through until the knot rests unseen between back of label and clothing
4. continue to sew (usually left to right if you are right handed) along the long length of label. sew over and over the edges of the label using the tiniest stitches possible that are not deep and are spaced evenly and close together. Make sure that as you sew, you also catch a thread or two of material of clothing underneath which is what will secure two together. Try not to let your needle go right through the clothing or you will have a label shaped rectangle of stitches showing on the outside of the garment.
5. you can use a slip stitch or an oversewing stitch as illustrated here
the slip stitch doesn't have to be two stitches together - one is good enough- the end result looks more like the oversewing stitch illustration
if you Google 'hand sewing' or 'basic sewing stitches' you will find lots of videos showing you the correct technique
6. when you are approaching one end of the label, fold the end over underneath itself (there is usually a line of different coloured thread to indicate where)and continue to sew around the fold
7. continue stitching neatly along all lengths and sides of label until you reach where you started. make a few small discreet stitches in same place underneath back of label and bring thread through and out the other side. cut off thread.
Also, don't sew nametapes onto elasticated areas, e.g. trousers/skirt waistband, as the stitches will rip when the elastic is stretched. Sew them onto the washing care label, it's lovely and silky to sew onto!
Milk I sew them on because once done, they don't rub off and everyone can see the clothes belong to my kid. I don't do it very neatly. DS still has a habit of coming home with other kids sweatshirts and their parents have done them in ink and to be honest I can only just read the name often I can't and they end up in the lost property box. He also belongs to a number of sports clubs so I do them as well as all the kids kit looks the same. Lose kit and uniform is not cost effective. This is not a bad skill to learn. I'm in a uniformed occupation and I use the labels on my own jackets so no one can say it isn't mine!