I have a year and a half old mini-strawberry (the small-fruit kind) plant. I have heard a lot about strawberries growing runners, but mine never did.
Is there something I should do to "coax" it to start?
Strawberries, or common strawberries, will produce runners if the conditions are right.
To coax runner production, increase nitrogen feeding after the harvest is over. High nitrogen lawn fertilizer will work for that. This is how we quickly reproduce nursery plants.
If your plants aren't producing runners at all, does check if the conditions are right. Are they drying out all the time? Is the soil nutrient level sufficient?
Some strawberry cultivars are much more vigorous runner growers, and some rather slow. June bearers are typically the more vigorous ones, and fall bearing and day neutral varieties produce fewer runners, sometimes almost none.
I have strawberries and they grow runners all the time. I don't think you can do anything to coax them to grow runners. They either do or don't have runners. That has been my experience so far.
As someone else mentioned, it depends a lot on the cultivar. Everbearing varieties are less likely to put out lots of runners. This year, picking a first crop of strawberries and diligently picking off shriveled or dying leaves as soon as they appeared seemed to push more runners into appearing. However, this didn't work as much for the smaller alpine plants (except one, which has gone completely haywire and produced runners all over the place).
We also just had loads of rain followed by a few weeks of very sunny weather. The runners started appearing by the end of the first week of sun.