לזכות רפואה שלימה לאדמו"ר ר' יצחק מנחם בן גיטל מירל בריינדל לאה
בתוך שאר חולי ישראל
The gemara says that first you put on your right shoe, then your left, tie your left and then tie your right.
Tosfot [I usually end that word with an "s" but in honor of the yahrtzeit of Eliezer Ben-Yehuda which falls out this year, I used the modern Israeli pronunciation] explains that we put on the right shoe first because we see that the right has more importance that the left for a meztora who has blood placed on his right ear, right foot and right hand. The left has more importance with respect to tying because we tie the tefillin on our left arms. [What about women who don't put on tefillin. Do they have to follow this order?? The poskim discuss this...]
The Bechor Shor asks what the halacha is for a lefty. He definitely should put on his right shoe first because we see the importance of the right from the metzora. The question is what shoe he should tie first. Since he ties tefillin on his right arm, maybe he should tie his right shoe first.
He discusses this issue at length and he suggests three possibilities why the left is tied first: A] We want to show the importance of the left side when we tie our shoes because we tie tefillin on our left arm. B] We want to tie on our weaker leg just as we put tefillin on our weaker side. C] Left is inherently more important with respect to tying, not only with respect to tefillin.
If this lefty also has a stronger left leg then the first two reasons would mandate that he should first tie his right shoe [because he ties his tefillin on his right and his right is weaker]. He concludes based on a diyuk in the gemara that if he is a lefty but his right leg is stronger he should first tie his left shoe like everyone else but if his left leg is stronger he should first tie his right shoe.
The Malbim differs and says that regardless of which leg is stronger a lefty ties his right shoe first.
For those who want to do more research, see the Shevet Hakehosi [4/1] and the Minchas Yitzchak [10/3].