I'm not sure this question can be answered "yes" or "no"; it really depends on what you think Mrs. Younger's motive is for giving Walter part of the inheritance money. Walter has not been successful in life, and Mrs. Younger knows this about her son. His lack of success or inability to achieve the American dream has emasculated him, and he no longer feels like a man who can provide for his family. Walter feels like this is his big break even though it involves some risk to go into business with Willy Harris, a somewhat shady character. Mrs. Younger understands this. She hopes that the business venture will be profitable and gives Walter the money to build his self-esteem. She hopes that he will be able to take his place in the family as the main breadwinner. Walter's current job is a chauffeur for a white man; this opportunity to own a liquor store will allow him to be his own man and not in service to another. Unfortunately, Walter's business deal is a bust. So, should Mrs. Younger have had faith in her son in hopes that he could achieve his dreams? If you think so, then she was smart to support her son. If Mrs. Younger was wrong in giving him the money because of Walter's past ventures, then the answer is she should have kept all of the money for the new house she wishes to buy.
It's a difficult decision for Mrs. Younger because she loves her son and wants to see him succeed. She gives in and supports her son simply because she would like to believe in him one more time.