There's bunch of words with
-en form of word like
brighten and others, but
weighten aren't valid pair. Is there any reasons, why is it like that? Is there any rule for that kind of morphology?
These "gh" spellings have germanic roots and come down to us from our Anglo-Saxon heritage. A common conversion from noun to verb is to add -en. so we have long, length and lengthen (make longer); strong, strength and strengthen; fear, fright and frighten. And so on. As Edwin Ashworth says, that does not mean that all such possibilities have been used. But many have: cheap to cheapen; stiff to stiffen; straight to straighten. Some just have not caught on, probably because not really needed. The verb weigh, for example, is already a verb, so why would we need a verb meaning make heavier? We often want to make things lighter, and, lo and behold, we have the verb lighten. The only non-germanic word that allows this treatment is to christen. Otherwise, stick to the Anglo-Saxon monosyllables.
If you really feel you need a verb for making something heavier, other than weight it down (more). There already exists the verb to weight.
If you think that the right word does not exist for you, you can coin a neologism, provided you are sure its meaning will be obvious enough, and signal what you are doing by placing it within single inverted commas.