There are two exceptions to this rule (or maybe even more but I don't remember other examples). 'Время' in the genitive case takes form времени; 'имя' in the genitive form takes form имени. У меня не былО времени. У него не было имени.

Время, племя, имя, вымя, семя. стремя... пламя. Знамя. временА, племенА, именА, семенА, стременА, знамЁна. Вымя, 'udder' and пламя, 'flame'/'blaze' don't have plural in literary Russian, but gen. sg. is вымени, пламени). Compare семя to Latin 'semen'. And consider Latin 'nomen' etc. There was a time - in proto-slavic epoch - when Slavic langauges started to look a bit like Japanese:) All the syllables bacame open. We lost that -s which you famously find in Latin -us and in Baltic languages. -en-/-em-/-in-/-im- before consonants and at the final position became nasal vowels whihc later became -я- (on/um has evolved into у, accordingly). But they never changed before a vowel. That's why we have понИМать/понЯть and other pairs with this root. So that is what happened with indo-european suffix -mn̥ You find it in both Latin and Russian semen. ANd we have this row of words as a reminder:) PS, I know, this is not exaclty what you asked about, but you said 'why':))))) Don't expect much regularity form our innflection paradigms.

I am not a native Russian speaker and in fact am learning the language myself. Previously, I assumed that the genitive form of "vremya" (sorry that I can't type in Cyrillic), was just irregular and just memorized it as "vremeneny." However, just today I found the following in a Wikipedia article about Russian grammar: "There is also a group of several irregular "different-declension nouns" (Russian: разносклоняемые существительные), consisting of a few neuter nouns ending in -мя (e.g. время "time") and one masculine noun путь "way". However, these nouns and their forms have sufficient similarity with feminine third declension nouns that some scholars such as Litnevskaya[8] consider them to be non-feminine forms of this declension, as written in the tables below." It would seem then that despite the "ya" ending, which typically indicates a feminine noun, "vremya" is in fact NEUTER and follows a different form, one apparently shared by a few other words, as KP notes below. Interesting. One still has to just memorize the different genitive ending but for me knowing that it is an unusual neuter word made it -- along with the words for "tribe" and "seed" for example (which also puzzled me previously) -- more understandable and thus readily remembered.

Just remember у меня нет времени(I does not have time). There is not "времи" word in russian language.

+ У меня ЕСТЬ (было, будет) время (Nom.case). - /absence of something (with negation)/ У меня НЕТ (НЕ было, НЕ будет) времени (Gen.case). Examples: — У тебя есть время для нашего разговора? — Извини. Сейчас я спешу. НЕТ времени. Поговорим позже, через полчаса. Хорошо? — Но потом я буду занят и у меня НЕ будет времени.