Would take a ribbon like grosgrain or a cotton (maybe something that don't ravel much) and embroider their names and then turn in on long sides and sew in or stitch all the way around. First link gives information like our community and second link is a visual video step by step. If all that don't work there is always a Sharpie Marker (but repeated washing could fade the marker).

Sandra, when I had to make name tags for my Mother (Miss Loretta) I created a set of them - in my hoop - then laid a ribbon along the stitching area - stitched away - then clipped between the names - I had allowed enough space - to fold the ribbon over on the edges - and tacked them down with my machine on each side - she loved them and they held up very well. What are you using for your names - when you zig zag around them - are they on material or ribbon - or what?

1. Sew with love - you can leave out the words and add your own

When my father-in-law entered a nursing home, I actually embroidered his name directly on the clothing -- it could be done on the facing of the garment & wouldn't show.
Blessings,
Diane

hi, I have a brother innovis4000 and the machine has inbuilt alphabets. I had entered the name on the screen and then on ribbon i embroidered the name . This does not need the hoop . It is on the sewing mode. I found it very easy

Here's a free ribbon embroidery tutorial AND a free set of guideline for placement of your ribbon. Seems informative. Good luck.

If you do not want to use a ribbon, try binding. Aside from the various sizes, you can use colors as well. I also have a Brother ULT, and it has the ability to embroider simply without hooping onto fabrics. Perhaps your machine can also do that? Then you can just embroider onto the pieces you want to put names on.