This refers to the scene in "D Minus Zero" where Harper is trying to fix the firing controls, if you look at his left hand he's wearing a wedding band.

You know I was pretty upset with myself at that point. I felt like I had let everybody down and that it was my fault that the Andromeda was damaged so badly that we had to hide. I had let those fighters get past me and Tyr had to bail me out and in doing so he got into trouble with Dylan. I wanted so badly to impress Tyr and to make Dylan proud of me but it didn't work out like I had planned.
I was always trying to impress people, that's what my Da told me. Mum said people should like me for who I was not for who they wanted me to be but it's a habit I've never been able to break.

There were only a few things I brought off Earth with me when I left with Beka and Bobby. Being an Uber slave didn't leave me with a whole lot of personal items, just a few small things that could be easily hidden, one of them was the wedding band my Da gave my Mum. When they died I kept her ring, it had been in the Harper family for six generations and I wasn't about to leave it for some grave robbing scum.

When I get upset I like to wear it, it helps me remember the sound of her voice and the touch of her hands. I feel like they are still with me, grounding me and helping me move forward when all I want to do is find a place to hide.
That's why I was wearing a ring when Dylan asked me to build the FMS.