With Glasgow being in a zone 9 you would have no difficulty keeping some greenery going. I suspect that part of the problem with the grass might be the shade of the tree. Once it has been removed the grass will certainly grow better. If you have your heart set on a courtyard garden it should not be hard to put in but most of those lovely pictures you see are of designed gardens and designers are not cheap.
Some of the things that have to be considered is the direction of the sun, some places in the courtyard will be quite sunny and some places quite shady. Drainage and the direction of waterflow from roof to where ever it leaves the property will have to be taken into consideration. The more paving the more the concern for what to do with water when it rains. This shows where water stands and moss grows as a result:
Pervious pavers, the ones Briolat mentioned, have become more common. They allow water to flow through and low growing plant materials like grasses and herbs (Luvmycat's thyme is one) to grow. Not everyone knows how to work with them so you would need expert planning, advice and/or landscapers to instal them as part of the makeover.
Of course you could catch rain water and turn it into a water feature like the following:
You can make a simple design of crossed walkways meeting at a central urn filled with plants and 4 plots of grass. It's modern and old all in one, should drain as well as what you have now and give you the option of creating 4 flower beds should you want to change in the future. Or you can use a combination of paving and stone to allow drainage somewhat like this:
What softens this look are the plants growing around the edges as well as the potted ones in the middle.
An ultra modern design you could easily imitate would be large slabs alternating with well mulched and amended ground holding a single plant or shrub like this:
A paved over courtyard needn't be drab or dull but it requires as much work as a standard garden would in the same space to keep it looking its best. I hope that's given you some of what you were looking for.