I had heard a theory a long time ago that Betty was Jameson's illigitimate daughter, and that Paul and Betty were Vicki's parents, making Liz the aunt of Vicki. Liz then sent monthly checks for Vicki's care out of guilt after she thought she killed Paul.
Yes, that was the theory I came up with along with Cassandra Blair, which Midnite has kindly provided a link for. This theory is a bit more elaborate than my original idea, which was a bit simpler but which also satisfies the evidence. The original idea was first suggested by a now-unknown DS fan named "Bob", although I believe I refined or changed it slightly. I'll try to reconstruct that scenario briefly based on memory and going over the post that Midnite has provided the link to.
Both my original theory and the more elaborate one with Cassandra Blair take into account the sketch of Betty Hanscomb and references to a butler at Collinwood named Hanscomb; to this I adduced possible inspiration from the 19th century Gothic mystery "The Woman in White," in which the unknown woman of mystery turns out to be the half-sister of the heroine. The name "Hanscomb" is the key here - it's not a very common name, and seems more than coincidental that the DS writers came up with this name in a mystery that simillarly concerns a young woman's unknown origins and her connection to a family of wealth and prestige.
We know that Elizabeth and Roger's father was Jamison Collins. In this scenario, Jamison later had an affair with the young daughter of his butler Hanscomb, named Betty. The affair produced Victoria. Hence Victoria Winters is a Collins and the younger half-sister of Elizabeth and Roger. Sam Evans' sketch of Betty clearly resembles Victoria. Sam most likely knew that Betty Hanscomb, who spent a lot of time at Collinwood, had had an illegitimate daughter and then died young. When Sam saw how much Victoria looked like his sketch of Betty, he probably guessed that she was Betty's daughter and that the father might have been Jamison Collins - especially knowing that Victoria had been brought to Collinwood by Elizabeth Collins Stoddard. I believe that all the servants at Collinwood had been fired at about the time of Betty Hanscomb's pregnancy obviously in an effort to stop gossip.
The more elaborate version of this scenario has a couple of things going for it that add interest but aren't really necessary. This version suggests that Jamison Collins had an affair with a daughter of his butler Hanscomb, and that this affair produced Betty Hanscomb (who obviously did not inherit the Collins name but her mother's name). Betty Hanscomb is the half sister of Elizabeth and Roger. Sam Evans' sketch of Betty Hanscomb not only looks like Victoria, but also Elizabeth, which is explained by this scenario. A further twist is that Victoria Winters was the product of an affair between Betty Hanscomb and Paul Stoddard. Elizabeth knew of the affair and of her husband's illegitimate daughter (Carolyn's half-sister) and made provisions to support her. Victoria Winters, then, was a descendant of Jamison Collins and the niece of Elizabeth Collins. Perhaps this more complex version is more in keeping with the convoluted relationships found in soap opera tradition.
Finally, let me offer this quotation from an interview with Dan Curtis, heard on one of the DVDs. Mr. Curtis was asked why the question of Victoria Winters' identity was never resolved. Mr. Curtis responded:
"We never got around to what Vicki's background was, and I never knew it."
I believe this accurately reflects the early days of writing and producing Dark Shadows. The writers would seem to have been given some guidelines and suggestions, but were free to develop their own ideas too. It's significant that all of the scripts using my ideas above are attributed to Francis Swann, who was also a novelist. In my opinion, it was largely in order to satisfy fans that DC decided to legitimize the idea that Elizabeth was Victoria's mother. I think that was one of the possibilities and may have been the suggestion Joan Bennett followed, but I think that was only one hint among many that was suggested in the early days of the mystery - and there were sure to be many suggestions and red herrings to keep viewers interested.