Originally, judo belts carried no adornment:
However, embroidery is a long established textile art in Japan, and as judo became more popular it was common for students to embroider their name on their judogi and belts to distinguish them at training:
It is ideal to carry the Judo-Gi in a bag, folded properly. To preclude loss, use small lettering or printing to mark your name on the lower left side lapel, upper trousers and belt.
This practice became something of a tradition in Japan, and hence was passed on to western students who trained there:
She was also presented with a handmade judogi from her principal teacher, Umetsu; Noritomi gave her a judo belt with her name embroidered upon it.
Eventually becoming something of a trend in western dojos also:
It has become popular to wear a belt with your name embroidered down one side and the organisation (judo, karate and so on) down the other. There are some firms that ...
It is not uncommon to see on belts with embroidered names the person's club/dojo's name also (on the opposite end). This is often more dependent on the tradition of the dojo itself however:
There is further use of the uniform to express group membership, especially in arts like judo and kendo, where large groups of practitioners get together for tournaments. Patches bearing the name of the dojo are frequently sewn onto uniforms and are both an expression of pride and membership.
Such patches are important expressions of allegiance even within the dojo itself. In my study, I noticed that new students generally did not wear such patches, while senior students did. Only when beginners became more involved with the dojo, grew more skillful, [sic] and in fact became more well integrated into the fabric of dojo society, did they generally opt to wear these badges of social identity. In the judo dojo I studied, for instance, it was really only after the first kyu promotion, when the student could exchange the white belt of a novice for a colored one, that he really began to feel an accepted part of the dojo. A short time after many students donned their first colored belt, they also began to proudly wear an embroidered patch whose Japanese characters spelled out the name of the dojo.
- Anthropos (Vol. 85, Issues 1-6) (p.60) (1990)
While traditionally dan grades were not distinguished by belt (asides from the black/red-white/red colour), tabs are sometimes added to signify the level of dan grade in Japan (similar to how mon are used in e.g. the British system for juniors). I suspect this is due to influence from Karate where this kind of marking is common.
Slightly before we start seeing names embroidered on belts in the west, we start seeing belts with the kanji for the dan grade embroidered on them (as opposed to a number of tabs):
According to current regulations, in IJF competitions belts (in addition to the mandatory "IJF Approved" logo) may only include one instance of the athlete's name on a 20x4cm patch at one end of the belt: