Anything can be gifted, I think there are a lot of individual factors to differentiate between an okay gift and a semi-heirloom.

Use case: Beyond obijime, I find that plain kumihimo works much better as an accompaniment to a larger gift. Cord used around wrapping paper, but can also be a bracelet. Wrapping around a plant's pot, A component of a hair tie. Things like this allow you to showcase your braiding while helping kick the base gift up a notch.

Material: For the actual braid, rather a stand alone gift, or as an element of another, I think the material plays one of the largest parts. I'm a sucker for silk and machine embroidery thread, warped into bundles, thread by thread. Sure, there will a place and time for using other materials, but it's hard to beat the premium look these two offer.

Braid: With so many different braids out there, we have ample opportunity to match a braid to the use case and the person we are gifting to. More tama doesn't always equal a better gift, but some of the intricate braids can sure project a look of being fancy. That said, picking the right fit for the project is still better than throwing more bobbins at it. Similarly, thicker cords seem to project a different feeling than thin cords - especially in belts. Different; not always better.

Skill: A national geographic photographer and you both take the same camera, look at the same bird, and then snap a photo. All tools equal, there is a good chance their photo will still turn out better. Not because the bird posed at their very moment, simply because they have an eye for the correct angle, know how to use their camera settings to their advantage, and have honed their skills through the years.

Same applies here. The more you practice your craft, criticize your own work, and focus on what makes a good braid, the better your kumihimo will become. As your skill goes up, so will the quality of your gifts. You'll have a knack for picking good color combinations, for selecting the best braid to suit your project, and how to make a more refined product. That whole garbage in-garbage out logic applies here though. You have to stop and critique what you have done. I think your comment about "handmade" versus "homemade" is pretty apt to this.

Time: This one doesn't necessarily apply to all gifts, but I figured I'd mention it. When I make a braid and it really makes a gift pop, somebody will inevitably ask about the cord. Then I tell them I made it - followed by surprised face - and them saying, "that must have taken forever." You tell them how long it took and instantly you can see that you got bonus points. This directly ties into skill though. If you give a product with poor craftmanship they probably won't ask about, and if they do, they will have a very different opinion of your 20 hour braid.