I'm glad to see this question, because I also have a family Bible that needs to be protected. This is the information I've found so far while researching how to protect our own Bible (I'm not an expert).
Cedar products are marketed to us to help guard against the intrusion of pests, but that doesn't mean the cedar is good for long-term storage of the things which are kept inside. Any wood product has oils and acids which could damage the contents inside a cedar box.
Your grandmother had the right idea when she wrapped the book in a cotton pillowcase. A cotton cloth which has been thoroughly washed would have given the book some protection without subjecting it to the kind of outgassing that happens with plastic or wood products.
The national professional association for conservators in the USA is the American Institute for Conservation of Historic and Artistic Works (AIC). They maintain an online directory to help you find a conservator in your area. I understand that you don't want to have the book restored, but if it would ease your mind to have a professional evaluate the book in person, and give you advice tailored to your own Bible, the AIC's directory would be a good way to find a specialist. Here is the link to their online aid to Find a Conservator.
Since 2010, the American Library Association has sponsored Preservation Week. Previous years' preservation week activities have included workshops at libraries, special sections on websites, and featured publications designed to help us preserve our collectibles.
Many of these publications suggest storing rare books in custom-fit enclosures. This protects the book against further wear and tear. Archival products suppliers sell enclosures made of fibreboard made for the purpose, that do not have acid or lignin which would harm your books. There are two styles: a clamshell box, intended for more permanent storage, and a phase box, a temporary enclosure designed to protect a box while it is waiting for the next step (or 'phase') of a conservation project. See the resources list below under 'box construction tips' -- the NEDCC leaflet (first in the list) has diagrams showing how to measure books for custom-made boxes.
Another thing to consider is the environment that your book is kept in. Old fragile books do best when kept away from extremes in temperature or humidity that you would find in attics or garages. One website suggests storing your Bible in an inside hall closet in order to keep it away from the exterior walls, and to protect it from being subjected to too much light. But the ideal spot will also have air circulation, and you need to be able to inspect your keepsakes for pests like silverfish or mice, and hazards like dust.
If you have a library or museum near you with a rare book collection, another thing you might consider is consulting the librarians to see what suppliers and methods they use. If you are in the US, ask your library if they participate in Preservation Week, and if they will have workshops or pamphlets on preservation available. 2015's Preservation Week is April 26th - May 2nd.
Box construction tips:
- Preserving Your Family Records: Conversation and Questions, by Mary Lynn Ritzenthaler -- Session 2 of NARA's Virtual Genealogy Fair 2015. handouts Video of Day 1 on Youtube
- an earlier edition of the talk Preserving Your Family Records, by Mary Lynn Ritzenthaler, from NARA's Virtual Genealogy Fair 2014 Video of Day 1 on Youtube
These recommendations are intended for guidance only. G&FH.SE does not assume responsibility or liability.