Avoiding clartiness probably means doing the hilly bits first and easier bits later. I'm assuming she is already a decent cyclist*. The WoL path is easy but loses height; I would tend towards staying on the main road, then down Gillespie Road, up Colinton village, up Redford Road, and thence Frogston Road and Captain's Road.
The Gilmerton climb is a bit wearing after then but diverting into the valley might be doable; round the back of the Morrisons, across Gilmerton Road, Mordun Park Street, Craigour Avenue and pop out opposite OPD6.
For a quieter route, I would agree with IWRATS' route as far as the Morningside clock. Cluny Gardens is well graded but it's busy.
Instead, turn left at the junction, then turn right at Canaan Lane, and go through the Astley Ainslie to South Oswald Road and St Alban's Road, South Lauder Road and Fountainhall Road. West Mayfield, Craigmillar Park, Lady Road (using the crossings at the railway roundabout if need be), and then Old Dalkeith Road the rest of the way. However, Lady Road to Cameron Toll to Inch Park and through to the bottom of Kingston Avenue would be even quieter, but requires an awkward right turn onto ODR.
* Edit: Pre-subsequent Gembo post suggesting otherwise.