Hi Marylou,
Did your friend ask in person at Horncastle library, because I find it a bit hard to believe that they don't have even filmed copies of the local paper(s). The library is supposed tp have a 'local/family history' section and newspapers are probably the major source of local history. (Actually, once you've got the film wound on to the machine the correct way

viewing papers on film is much easier than struggling to read the old broadsheet ones when they're bound in ledgers.)
If your friend can get to the library then she can also look through the other local papers for that/nearby week(s) to see if there are any other articles about him. Just depends exactly why he was in the paper.
Horncastle library details
http://www.lincolnshire.gov.uk/libraryDetails.asp?library=NE1&catId=2434&multimap=1Other Horncastle papers
http://tinyurl.com/23vc2bIf your friend can't get to the library the staff should be able to provide her with a photocopy of the article, though she may have to pay a small charge.
Depending on what the article says (might be he's already married/engaged) then it's a case of searching through the GRO Index till she finds some likely candidates. If the number of BMDs for William Laming between 1901 and 1920 are any guide the there won't be too many marriages, though she will have to also search the deaths as well.
Has your friend sent for William's birth certificate yet? If so, she will have his mother's maiden name and from that should be able to find siblings on the GRO Index. Sometimes tracing siblings and their descendants can help to find the person you're really looking for.
The online BT phone directory has a Laming in the Horncastle area who may or may not be connected but I would strongly advise her to write and not phone.