In "Hamlet" what might Queen Gertrude's life been like if she had refused Claudius's offer?

What an interesting question!  She would have probably have been lonely; Hamlet's descriptions of her seem to paint her as a woman who like to be in company, liked to have someone around.  So, she would have been lonely.  She also would probably have been a lot more stressed; many more responsibilities of the kingdom would have fallen on her shoulders, and it would have been quite a strain.  She also would have had a better relationship with Hamlet, and possibly had the opportunity to oversee his wedding to Ophelia.  It is because of Hamlet's fury at his mother for marrying Claudius that he alienates himself from her, lectures her about fidelity, and is caustic and rude to her.  Because of her seeming infidelity, Hamlet also rejects Ophelia, who, by all counts, he had been very honorably courting, with serious and good intentions.  But, because he was so angry with women, so betrayed by them, and found them so lacking in trust, loyalty and prudence, he rejected Ophelia outright.  Who knows what might have happened between them if that anger hadn't been there; so, Gertrude might have become a happy mother-in-law to Ophelia.  So, there would have been loneliness and stress, but balanced by a good relationship with her son, and happy times connected to that.