I liked Lincoln's Repeater in Fallout 3, but never loved it.
I never actually used This Machine much on account of not knowing what to use it for.
If I had to choose between the two, I'd pick the Repeater. If I had to choose what I personally think the combustion mascot of New Vegas is I'd pick Lucky.
This. I LOVE the way Lucky is introduced. No one's carrying it, no note or backstory is attached to it; it's just there. Other character's have their signature weapon; Graham has ALSID, Ulysses has Old Glory, but Lucky? It's the Courier's weapon. It's yours to use.
Lucky definitely gets my vote for being the mascot weapon of Vegas, and it's surprisingly powerful in the right hands (saved my ASS in Old World Blues, 100% sure I would've died had I not taken Lucky) despite having such a modest base damage.
Lincoln's Repeater seems like the mascot weapon of Fallout 3 cause it's such a patriotic, powerful and legendary weapon. New Vegas? Lucky fits the theme of Vegas perfectly and has no backstory or previous owner, suggesting it's intended to be the Courier's signature weapon, imo.