The door of your observatory will get plenty of use. I suggest a solid-core door with good hinges, a tough lock, and a good seal. To improve security, add a dead-bolt.
Some amateur astronomers who live in cold climates have attached a "warm room" to their observatories. This idea is more practical when the observatory is far from home. For a backyard observatory, you can always dart into your (darkened) house to warm up.
If you are planning a warm room, be sure to insulate the wall between it and the observatory. Heat currents are the bane of seeing (image steadiness), and even the warm room's roof will give off enough heat, when the room is in use, to seriously degrade any images in the sky above it.
The most important interior consideration is power. Alternating current must be run to the observatory. Whether you install an electrical sub-panel in the observatory (the best option) or run a line from your house, allow for at least two circuits, each with 15-amp capacity. If you're adding a warm room, add two additional circuits, and one of them should provide enough amperage to operate whatever model heater you plan to install.
Lighting is the second interior consideration. Dimmable red lights are mandatory. Dimmable white lights are optional.
Next is ventilation. If your observatory is sealed too tightly, you will experience heat-retention problems no matter how well you made your roof and outer walls. Some amateurs use passive ventilation, while others use quiet, low-flow fans.
If you have the space, do yourself a big favor and build or purchase some storage in the form of shelves or cabinets. If space is critical, consider a rolling tool cabinet. If you choose a roll-off roof over a rectangular building, don't neglect the room's corner spaces, which are great for storage. Plan some space around your telescope(s) for observing chair(s). Pegboard panels allow for extra storage without taking up floor space. You'll be amazed at how many items you can hang on hooks.
Finally, be sure you install a clock. I suggest one with large numerals, illuminated by red light-emitting diodes (LEDs).
I'll finish this series next week by talking about your telescope's pier.