Lisa,

Welcome! 

You could try adding the name to your image, experimenting with a size of lettering that works well with it, and is still readable.

If that doesn't work for you, if the original looks better without the name, you might want to make a separate chart for the name only, in a font that will show up well and be a good size, and then perhaps cross stitch it on after the blanket is done. (For that, I'd then also suggest a practice letter or three on a square or rectangle of crochet, to see if you like how it looks.)

If you don't like the way that looks, you could embroider the name onto the finished blanket, using anything from a very simple backstitch or simple chain stitch (or French knots - takes more patience than I'd have, lol, but it can really look lovely!), perhaps in a larger weight fiber (e.g., a bulky on a worsted) to make the name stand out more, or directly chain stitch with your crochet hook onto the finished blanket.

Anything you're unsure of, I would suggest trying on a simple crochet swatch, to see if the technique is comfortable for you and looks as you're picturing it in your mind. Of course, you don't want it to become a chore, because imo, half the fun of it is the enjoyment.  Luckily, there are a lot of excellent options, and especially, if you have a pattern that's turned out the way you already like it, no need to reinvent the wheel to get the name into your pattern for your lucky recipient.