In the Convert Lead page, the Campaign Source field lists only the campaigns to which the selected lead (that is being converted) is associated. It does not list all the campaign records available in your Campaigns module. So please check if the lead is associated to any campaign.
A lead could be associated to more than one campaign. In such cases, when you convert the lead, the Campaign Source field is useful in selecting the campaign which qualified the lead to a deal.
After lead conversion, the Contact record will have all the campaigns that were associated to the lead, while, the Deal record will have only the campaign source that you selected during lead conversion.