I will be in Bangkok for a week and would like to rent a bicycle to ride around Lumphini park. Even if I rent for a few hours or a day.

Possible?

I suspect you can rent a bike right in Lumpini park somewhere. If not, there is a bike rental stall near Chidlom BTS I've seen on the street there, which is not too far from Lumpini.

* there's also a bike shop directly across from the park north on Sarasin called Probike. They have a website. They might rent bikes.

You can also go along rama 4 to samsen mrt, there is a rental there, outside chantchuri square or whatever its called.