You didn't say where you are. It makes a difference, in what you would do.
When would be NOW if it is fall where you are. Collecting seed from them depends on the neighbor leaving the flowers on to set seed. People that harvest their lavender don't do that, they harvest the flower spikes as buds, before it ever sets seeds. So first you would have to ask the neighbor if she would save a couple flower spikes, leave them on to set seed.
The flower spikes need to stay on the plant until all the flower petals have dropped off. Then you can harvest them, let them dry a little and rub them over a strainer. The seeds should fall through. Collect them and put them in a Ziploc baggie with moist potting soil and put them in the fridge for three weeks. This is called cold stratifying and it simulates going through winter. Alternatively if your winters aren't too cold and it is fall where you are, you can collect the seeds and plant them in the ground now to overwinter. It's a bit more of a gamble since you can't provide appropriate conditions that way (will they dry out? will they get too wet and rot out? etc), but it is what nature does. If starting indoors, after the cold stratification, plant the seeds in potting soil in a warm place under lights. The seeds need lights for germination, so don't cover them, just put them on the soil and press lightly to be sure they are in good contact with the soil. Keep lights directly over them, just a few inches above on for 16 hrs a day (see the seed starting basics thread in this section). For warmth, best is a heat mat under the tray, for gentle even bottom warmth.
Keep the soil just damp until the seeds sprout, anywhere from 2 to 4 weeks. Then tend the little seedlings carefully until they are transplant size, which will be 6 - 10 more weeks (they grow slowly). Then they can be transplanted. They probably won't bloom the first year from seed.
This is why imafan suggested cuttings!