Именительный - from имя (name)
Родительный - from родить (to give birth), and as Demonic_Duck mentioned all related words fit here too.
Дательный - from дать (to give)
Винительный - from винить (to blame, to accuse)
Творительный - from творить (to create)
Предложный - from предлог (preposition)
Latin (or English) names of cases mean pretty much the same.
You may also find useful this explanation from "Russian for Dummies":
In a Russian sentence, every noun, pronoun, and adjective takes a different ending depending on the case it’s in. What’s a case? In simple terms, cases are sets of endings that words take to indicate their function and relationship to other words in the sentence. If you’ve studied languages such as Latin or German, you know that different languages have different numbers of cases.
Russian has six cases, which isn’t that bad compared to Finnish, which has fifteen!
Nominative case / Именительный падеж
A noun (or a pronoun or an adjective) always appears in the nominative case in an English-Russian dictionary. Its main function is to indicate the subject of the sentence.
As a rule, the subject behaves the same way in Russian as it does in English. It answers the question “Who or what is performing the action?” For example, in the sentence "Бренда изучает русский язык" (Brenda studies Russian), the word Бренда (Brenda), indicating a woman who (like yourself) studies Russian, is the subject of the sentence and consequently is used in the nominative case.
Genitive case / Родительный падеж
You usually use the genitive case to indicate possession. It answers the question “Whose?” In the phrase "книга Анны" (Anna’s book), Anna is in the genitive case (Анны) because she’s the book’s owner.
Genitive case also is used to indicate an absence of somebody or something when you combine it with the word "нет" (no/not), as in "Здесь нет книги" (There’s no book here). "Книги" (book) is in the genitive case because the book’s absence is at issue.
Russian uses genitive case after many common prepositions, such as около (near), у (by, by the side of), мимо (past), из (out of), вместо (instead of), and без (without).
Accusative case / Винительный падеж
The accusative case mainly indicates a direct object, which is the object of the action of the verb in a sentence. For example, in the sentence "Я люблю русский язык" (I love Russian), the phrase "русский язык" is in the accusative case because it’s the direct object.
Some frequently used verbs like читать (to read), видеть (to see), слышать (to hear), and изучать (to study) take the accusative case. Like in English, these verbs always take direct objects.
The accusative case is also required in sentences containing verbs of motion, which indicate destination of movement. For instance, if you want to announce to your family that you’re going to Россия (Russia), Россия takes the form of the accusative case, which is "Россию".
You also use the accusative case after certain prepositions, such as про (about) and через (through).
Dative case / Дательный падеж
Use the dative case to indicate an indirect object, which is the person or thing toward whom the action in a sentence is directed. For example, in the sentence "Я дал учителю сочинение (I gave the teacher my essay), учителю (teacher) is in the dative case because it’s the indirect object. (“My essay” acts as the direct object, which we cover in the previous section.)
Some frequently used verbs, such as "помогать" (to help) and "позвонить" (to call), force the nouns that come after them into the dative case. The implication with these verbs in Russian is that you’re giving help or making a call to somebody, which suggests an indirect receiver of the action of the verb.
Instrumental case
As the name suggests, the instrumental case is often used to indicate the instrument that assists in the carrying out of an action. So, when you say that you’re writing a letter with a "ручка" (pen), you have to put "ручка" into the instrumental case, which is "ручкой".
Use the instrumental case after certain prepositions such as с (with), между (between), над (over), под (below), and перед (in front of).
Prepositional case
Prepositional case got its name because it’s used only after certain prepositions. Older Russian textbooks often refer to it as the locative case, because it often indicates the location where the action takes place. No wonder it’s used with the prepositions в (in) and на (on).
The prepositional case is also used after the prepositions o (about) and об (about). So when you say to that special someone, “I am constantly thinking about you,” make sure to put "ты" (you; informal singular) in the prepositional case, which is "тебе": Я постоянно думаю о тебе.