Give the glove a good cleaning and a light conditioning before you begin.
Don't completely unlace the glove. As you unlace, follow it with the new lace so you can retain the pattern. Some transitions are not obvious when webs come together with a finger or thumb. Some lacing patterns on the web itself can be confusing. Also, I like to cut my lace to a point instead of leaving it squared off when feeding it through the holes it won't jam or be as difficult when stuffing gets in the way. You will trim the end anyway later to however you like it to look.
Don't be afraid to lace it fairly snug, as the lace will relax with use. If you don't condition the lace (which I do not recommend), you should not pull it as snug because it can break under the stress of the first hard throws. By hard, I mean upper 70s and beyond. It really won't matter if you are re-lacing for someone younger than HS. As you run the lacing, be sure it doesn't become twisted. The lace has a smooth and a rough side, plan that out so you get a consistent look, or you will have to do it over. For example, you don't want the rough side in the pocket--it looks crappy.
I also put just a bit of conditioner on the lace before using it so it's less likely to snap soon after the re-lace. Also, most of the time, the laces break in unexposed areas or areas near the eyelets because that lace is the driest and often the least conditioned part of the lace.
Don't let your teammates know you did the re-lace, or you will be lacing gloves all season.