A couple of possiblities listed below....I'm thinking that it is the annual jewgrass.

Here is a link that might be useful: {{gwi:1349915}}

It looks like weeds we get here in rural Illinois. I don't know the name of it, but it does have some pretty blue blooms. My mom used to fuss whenever I pulled them.
Is it a weed or is it a "wildflower".?? LOL

Lucia

Lucia,

To use the old gardeners saying. A weed is a plant where you don't want it. Your mother liked them. I find them a good ground cover that is easily pulled or clipped when they get too big for the space. They fill in nicely the area allowing the spring bulbs to go dormant with out looking a mess. If you put your hand under the plant the soil there is generally cooler than under surrounding plants??? It is up to you to decide whether it is a weed for you or not.

As you can tell by the above I like your mother like the plant.

Lucia; you might be thinking of something in the commelina family, dayflower. Very pretty, but extremely aggressive, in full sun.

From the picture it looks like a common weed in my area known as Nepalese Browntop. If you find a way to kill it, please let me know.