There are multiple issues in the question to address separately:
The correct form
The only correct sentence is:
Das ist eines (nom. sg. n) meiner Lieblingslieder (gen. pl. n) von einer (dat. sg. f) meiner Lieblingsbands (gen. pl. f).
With Lieblingssongs, it would be:
Das ist einer (nom. sg. m) meiner Lieblingssongs (gen. pl. m) von einer (dat. sg. f) meiner Lieblingsbands (gen. pl. f).
Why isn't it eines meines or eine meine?
Because einer/eine/eines (indefinite pronoun) and meiner (possesive determiner, i.e. 'article' of the following noun) do not have the same case.
Einer/eine/eines acts like a noun itself and takes the case and number according to its role in the sentence, while meiner is dependent and congruent to its noun. (I guess you were expecting that they would both depend on the noun and hence would have the same case and number as well as the same ending.)
However, because of their semantical relationship, einer/eine/eines and the noun do have the same gender.
- The first pronoun eines is the head of the nominal phrase in the complex predicate with sein. Thus, it is nominative singular. It has the meaning of one in English. It matches die Lieblingslieder (n. pl; sg.: das Lied) in being neuter, or, respectively, die Lieblingssongs (m. pl; sg.: der Song) in being masculine.
- The case of the second pronoun einer is governed by the preposition von and thus dative singular. It matches die Bands (f. pl.; sg.: die Band) in being feminine.
- The phrases meiner Lieblingslieder and meiner Lieblingsbands are genitive plural. The genitive has the same meaning as the preposition of in English (one of my favourite …). In plural, all genders have the same endings; nevertheless meiner Lieblingslieder is neuter and meiner Lieblingsbands is feminine.
Why isn't it die Lieblingsbände(r)?
Because the plural of die Band is die Bands. This was already answered extensively by tofro. Der Band and das Band on the one hand and die Band on the other are homographs in nominative singular, but they are inclined and pronounced differently:
- Der Band ('volume', pl. die Bände) and das Band ('ribbon', pl. die Bänder) are actually spoken differently in singular and plural. In singular, their root features an [a], while the plural is marked by the umlaut [ɛ] (similiar to Engl. man – men).
- Die Band ('band', pl. die Bands), however, is a loanword; the orthography and pronounciation was adopted from English. Thus, it already has the umlaut [ɛ] in the singular root which is not marked orthographically. This means: the [ɛ] doesn't mark the plural and there would be no point to writing it with <ä> only in plural (that makes it different from man – men or Mutter – Mütter where the umlaut is the (only) plural marker and also reflected orthographically). Furthermore, -er is an untypical feminine plural ending (die Mütter, f. pl., being one notable exception).