I remember in Nürnburg people sometimes answering "bisschen" like "ein bissel", oder "ein bissler". Was this Bavarian or Berlin dialect?
In a nutshell
One of the variations you guessed (»bissler«) is wrong. What you heard was either »bisserl« or »bissel«. In Bavaria both are common variations of »bisschen«. In German German both words are non-standard words (i.e. dialect). In Austrian German »bisserl« is part of the standard vocabulary (i.e. not a dialect word). »Bissel« is just transforming from dialect to standard in Austrian German.
The suffix »…chen« in »bisschen« is marking a diminutive. You also find it in:
Das Kleid -> das Kleidchen (small dress)
Der Hund -> das Hündchen (small dog)
Die Pflanze -> das Pflänzchen (small plant)
Der Biss -> das Bisschen (small bite)
A bite (ein Biss) is the amount of food, that you can bite of at once. But you can bite of amounts of different sizes, and if you do a small bite, then this is in German »ein Bisschen«.
So the origin of the word was a noun, that was used like a unit to measure the size of something. Its original (but now obsolete) usage was like this:
Ein Liter Milch - a liter milk.
Ein Pfund Mehl - a pound of flour.
Ein Bisschen Hefe - a small bite of yeast.
But the usage as a noun became distinct, and the word became an indefinite pronoun. Now it is used the same way as »wenig« (some):
Ein wenig Hefe = ein bisschen Hefe = some yeast.
other diminutive markers
But »…chen« is not the only one diminutive marker in German. Another one is »…lein«:
Das Kleid -> das Kleidlein (small dress)
Der Hund -> das Hündlein (small dog)
Die Pflanze -> das Pflänzlein (small plant)
You can find »…chen« and »…lein« across the hole German sprachraum, but German is a pluricentric language, this means, that there are regional variations. And one of this variations are additional diminutive markers in the southern regions of German sprachraum. One, that is used also in standard German (i.e. not only in dialect) is »…erl«:
Das Kleid -> das Kleiderl (small dress)
Der Hund -> das Hunderl (small dog)
Die Pflanze -> das Pflanzerl (small plant)
None of this three examples is standard German, but the next are standard (i.e. not dialect) in Austrian German:
Der Sack -> das Sackerl (small bag)
(Der Häfen) -> das Häferl (small pot, mug)
The word »der Häfen« (the pot) is not standard German, this really is dialect, but its diminutive »Häferl« is a word of standard Austrian German.
And so in southern regions of German sprachraum (Bavaria and Austria) you also find »bisserl« instead of »bisschen«.
The status of »bisserl« in German German is: Dialect (this is true for all words ending with the diminutive marker »…erl«). In Austrian German it is a word of the standard vocabulary. In Austrian German you can choose between »bisschen« and »bisserl«.
hard to pronounce
Many people, not living in southern parts of German sprachraum have problems with the proper pronunciation of the diminutive suffix »…erl« which is [ɐɫ] (bisserl = [ˈbɪsaɫ]). Most people from norhtern parts either say [ˈbɪsɐl] (wrong l) or even [ˈbɪsl̩] (no /ɐ/ and wrong l). But the writing of [ˈbɪsl̩] is »bissel« (like »bisserl« but without r).
And what happened with »bisserl« happens with (almost) all words with the diminutive suffix »…erl«: They are pronounced in northern parts of German as if the suffix was »…el«.
bisserl > bissel
But since the word »bisserl« is one of the few words from the south, that just is spreading across the hole German sprachraum, it turns into »bissel« in the north, and in this form it slowly comes back to the south. So now, at the beginning of 21th century, »bissel« is also very common in Austria. This was not the fact at the end of 20th century.