If the mulberry tree is large enough - the heart wood will be something like golden teak in color. At least the Japanese mulberry wood is. I have a slab of mulberry and it is beautiful for turning bowls. I almost did not recognize it at the local home center because instead of turning gray when it weathers, it turns chocolate brown - except for the sap wood. A good finish will greatly slow the wood from turning.
I asked the fellows where I bought it what kind of wood it was and they said "looks like elm to me". I said "but it is brown." They replied "Well, it is a different kind of Elm." I finally figured out that it was mulberry by seeing that wood and color in two other stores - and the name is Kuwa (mulberry).
I also have a mulberry tree in my yard. It is messy but LOML made some mulberry jam for me.
For a pen, it depends on the grain pattern and the cut. It is always great to have something unique around. If I were you, I would save some wood for bowls, plates or vases. I have made one 10 inch diameter 2 1/2 in deep bowl from it and LOML loves it.