The German word meinetwegen is used in several different contexts. Its first meaning can be translated as "on my account" or "for me" in English.
Ich weiß, ihr seid nicht meinetwegen gekommen.
I know you haven't come because of me.
Caption 3, Verstehen Sie Spaß? - Sascha Grammel
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Hat Themba wirklich nur meinetwegen ein Eigentor gemacht?
Did Themba really score a goal against his own team just because of me?
Caption 33, Die Pfefferkörner - Eigentor - Part 6
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It can also mean "for example" or "assuming that":
Und wenn du dich jetzt meinetwegen dort auf die Seite drehst...
And if you now turn onto your side, for example...
Caption 63, Die Schmerzspezialisten - Diese Schlafposition solltest du unbedingt vermeiden!
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Additionally, meinetwegen can also suggest indifference, translated as "for all I care" or even the very rude "whatever" below.
Machen Sie doch meinetwegen, was Sie wollen!
But do what you want, for all I care!
Caption 61, Märchen - Sagenhaft - Der Rattenfänger von Hameln
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„Meinetwegen“, brummte Frederick.
"Whatever," grumbled Frederick.
Caption 42, Piggeldy und Frederick - Unendlichkeit
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Von mir aus is a colloquial expression, and can be translated as "for all I care," "as far as I'm concerned," or even simply "I don't mind."
In welcher Form auch immer... bis morgen früh von mir aus.
In whichever form… till early morning, as far as I'm concerned.
Caption 45, Fastnacht - Karneval - Quartier Latin
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Von mir aus bleibe ich ewig
I don't mind remaining forever
Caption 7, Max Giesinger - Nicht so schnell
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Was meinst du? -Von mir aus.
What do you think? -I don't mind.
-Vielleicht hat hier jemand was gesehen.
-Maybe someone here saw something.
Caption 43, Die Pfefferkörner - Eigentor - Part 2
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Further Learning
Take a look for additional examples on Yabla German and try working on your own sentences with meinetwegen and responses with von mir aus.