Keep the existing architecture. Don't plan on increasing the ceiling height or turning angled walls into straight ones. These expensive structural changes will add big bucks to the cost of your conversion and can easily be skipped for a child's playroom. Children will love the angles and secret places.
Add good insulation. Use spray foam insulation that will naturally conform as your home expands during the seasons. This will cost a little more upfront, but it will decrease your energy usage as well as add an important sound barrier between the noisy playroom and the rest of the house. If the insulation works well enough, you may be able to avoid adding HVAC equipment to the attic.
Use cheap flooring. If your flooring is too thick, it can affect the attic's insulation. Skip the hardwood floors, and go for a cheaper carpet alternative. Check with a professional to determine whether you will need to reinforce your floors with joists before installing your carpeting.
Consider whether you need to add electricity to the attic. If your attic has a window and your children will only play in the playroom during daylight hours, you may have enough natural light that you may not need electricity. The addition of a couple of battery-operated lights might provide any extra light you need as well as save you money by not hiring an electrician.
Use cheaper alternatives for your heating and cooling needs. An oil-filled heater can provide heat to the room but has a low surface temperature that is safe to use around kids.
Paint the walls with chalkboard paint. This will provide your children with hours of cheap entertainment and will prevent you from having to purchase a lot of toys to fill the space. String a laundry line on the wall, and hang your children's art from it to create cheap wall art.
Repurpose underused furnishings and materials around the house as pieces for your playroom. Place the unused recliner in the room. An unused bookshelf can store children's books and toys. An underused table can be your child's art table. Place attractive toy collections on shelves for decorations.
Place buckets of toys in a recessed area in your attic, and cover with a curtain. This will save you the cost of built-in shelving units or expensive storage devices.