Why is Amanda obsessed with finding a suitor for Laura?

As an aging mother, Amanda realizes that she needs to have her children in stable positions so that they may be able to provide for her.  Her conflicts with Tom indicate that she does not trust him to provide for her--she has also seen his application to the Merchant Marine. In addition, she may worry that he may follow the path of escape that her husband, whose smiling portrait lurks over the family. 

With the play's setting in the 1930s, Amanda certainly is aware of the limited opportunities for young woman.  This is why she has encouraged Amanda to enroll in Rubicam's Business College.  Then, when she discovers that Laura has dropped out of the school, Amanda sees a "gentleman caller" as the last savior for Laura.  This savior image is suggested by Tennessee Williams's production notes that the news from Tom that a man is coming is referred to as the "Annunciation" and the candles are Laura's "altar."

If Laura marries, Laura will be provided for, and Amanda gains some comfort from knowing that she has two people who can care for her in her old age.  Therefore, to Amanda, Laura's marriage is a solution to the family's problems.