I was able to find an authoritative source indicating a relationship between modern Finnish and Hungarian:

Hung. k é z (hand) = Finn. k ä s i , Hung. v é r (blood) = Finn. v e r i, Hung. m é z (honey) = Finn. m e s i, Hung. s z a r v (horn) = Finn. s a r v i, Hung. v a j (butter) = Finn. v o i, Hung. e l e v e n (alive) = Finn. e l ä v ä, Hung. m e n n i (to go) = Finn. m e n n ä, Hung. r e p e d (to be torn) = Finn. r e p e ä ä etc.. which give a direct hint to a common origin. To notice similarities between Hung. f e j (head) = Finn. p ä ä, Hung. f é s z e k (nest) = Finn. p e s ä, Hung. f é l (to be afraid) = Finn. p e l k ä ä, Hung. f a k a d (to become fulfilled) = Finn. p a k a h t u a and other words is considerably more difficult, if you are not aware that the letter f in the beginning of the word regularly match the Finnish p. Or, the letter n in Finnish is often replaced by ny in Hungarian, as in Finn. n i e l l ä (swallow) = Hung. n y e l n i, Finn. m i n i ä (daughter-in-law) = Hung. m e n y. The long ő, met in the end of a Hungarian word, has previously been a diphtong öü or eü and even more previously ev. The consonant v in this is still often met in words like, e.g. Hung. k ő [the accusative case k ö v e t ] (stone) = Finn. k i v i , Hung. t ő (tree base) = Finn. t y v i and Hung. v ő (son-in-law) = Finn. v ä v y.

But, more interestingly, Dani was right about Estonian. Estonian is much more closely related to Finnish. The Finno-Ugric group contains 7 languages. Except for Finnish at one end and Hungarian at the other, the 5 remaining languages are obscure ones that you or I have probably never heard of.

Aleck