C#: naming rules for protected members fields

You should not use fields that are protected, for the reason that versioning and access cannot be guarded. See the Field Design guidelines. Change your field to a property, which will also force you to change to name (as you cannot have two properties with the same name). Or, if possible, make the protected field private.

To make setting your property accessible only to the inheriting classes, use a protected setter:

public class Dimension : Text
{
    private string _textPrefix;

    private double _absoluteDistance;

    /// <summary>
    /// Gets the real measured distance.
    /// </summary>
    public double Distance
    {
        get { return _absoluteDistance  }
        protected set { _absoluteDistance = Math.Abs(distance); }
    }
}

Although that does cause divergence between get and set, as functionality is not the same. Perhaps a separate protected method would be better in this case:

public class Dimension : Text
{
    private string _textPrefix;

    /// <summary>
    /// Gets the real measured distance.
    /// </summary>
    public double Distance { get; private set; }

    protected void SetAbsoluteDistance(double distance)
    {
        Distance = Math.Abs(distance);
    }
}

Well, summarizing of what already being said you can do something like this :

public class Dimension : Text
{
    private string _textPrefix;

    private double _rawDistance;

    /// <summary>
    /// Gets the real measured distance.
    /// </summary>
    public double AbsoluteDistance
    {
        get; private set;
    }

    /// <summary>
    /// Gets the raw distance
    /// </summary>
    public double RawDistance
    {
        get { return _rawDistance; }
        protected set { _rawDistance = value; AbsoluteDistance = Math.Abs(value); }
    }
}

When RawDistance's value is set it also sets value for AbsoluteDistance and because of that there is no need to invoke Math.Abs() in getter of "AbsoluteDistance".