I am a lefty (when it comes to writing and using a fork), and regarding tefillin I was told by my Rav (way back when) there are 3 factors…
1) In general putting tefillin on the left hand is, as a general rule, the preferred way.
2) You put Tefillin on the weaker hand.
3) You put tefillin on the hand you do NOT write with.
In my case since #1 and #2 apply, i.e. according to 2 out of the 3 determining factors I should wear on the left hand, I do wear tefillin on the left hand (the way righties do).