Web part, you configure it on a template (under the Design tab), the property setting ties to the template.
Widget, you configure under the Page tab, the setting is like data/text you entered directly to the page, it's not stored in the template.
Those are two different things in Kentico. (Although widget is build based on web part)
So in your case, if you are saying it's web part, not widget, then the setting for the web part is going to be in the CMS_PageTemplate table [PageTemplateWebParts]
In that filed, you would fine web part property setup on a template been stored like this
<webpart controlid="Repeater" guid="acf93fb1-6f81-4769-aa7d-b917172ec6bd" type="repeater" v="1"><property name="alternatingtransformationname"></property><property name="backnextlocation">split</property><property name="cachedependencies">##DEFAULT##
</property><property name="cacheitemname"></property><property name="cacheminutes"></property><property name="cat_open_ajax"></property><property name="cat_open_output_filter"></property><property name="cat_open_performance"></property><property name="cat_open_time zones"></property><property name="cat_open_visibility"></property><property name="categoryname"></property><property name="checkpermissions">False</property><property name="classnames">DancingGoat.Office</property><property name="columns">OfficeEmail, OfficeCity, OfficeZipCode, OfficePhone, OfficeCodeName, OfficeStreet, OfficeName, OfficeCountry</property><property name="combinewithdefaultculture"></property><property name="container"></property><property name="containercssclass"></property><property name="containercustomcontent"></property><property name="containerhideonsubpages">False</property><property name="containertitle"></property><property name="contentafter"></property><property name="contentbefore"></property><property name="controlid">Repeater</property><property name="culturecode"></property><property name="customtimezone"></property><property name="datasourcename"></property><property name="disablemacros">False</property><property name="disableviewstate">False</property><property name="displaytoroles"></property><property name="enableoutputfilter">False</property><property name="enablepaging">False</property><property name="filtername"></property><property name="filteroutduplicates">False</property><property name="hidecontrolforzerorows">True</property><property name="hideonsubpages">False</property><property name="instanceguid">acf93fb1-6f81-4769-aa7d-b917172ec6bd</property><property name="itemseparator"></property><property name="loadpagesindividually">False</property><property name="maxrelativelevel">-1</property><property name="nestedcontrolsid"></property><property name="newbuttontext"></property><property name="orderby"></property><property name="outputconverttablestodivs">none</property><property name="outputfixattributes">True</property><property name="outputfixhtml5">True</property><property name="outputfixjavascript">True</property><property name="outputfixlowercase">True</property><property name="outputfixselfclose">True</property><property name="outputresolveurls">True</property><property name="pagenumbersseparator">-</property><property name="pagerhtmlafter"></property><property name="pagerhtmlbefore"></property><property name="pagerposition">bottom</property><property name="pagesize">10</property><property name="pagingmode">querystring</property><property name="partialcachedependencies">##DEFAULT##
</property><property name="partialcacheminutes"></property><property name="path">/Contact/%</property><property name="preservepartialcacheonpostback">False</property><property name="querystringkey"></property><property name="relatednodeisontheleftside">False</property><property name="relationshipname">isrelatedto</property><property name="relationshipwithnodeguid">00000000-0000-0000-0000-000000000000</property><property name="resultsposition">top</property><property name="selecteditemtransformationname"></property><property name="selectonlypublished">True</property><property name="selecttopn"></property><property name="showdeletebutton">False</property><property name="showeditbutton">False</property><property name="showeditdeletebuttons">False</property><property name="showfirstlast">True</property><property name="showfordocumenttypes"></property><property name="shownewbutton">False</property><property name="sitename"></property><property name="timezonetype">inherit</property><property name="transformationname">DancingGoat.Office.FooterOfficeContact</property><property name="useupdatepanel">False</property><property name="visible">True</property><property name="webparttitle">OfficeContacts</property><property name="webparttype">repeater</property><property name="wherecondition"></property><property name="zerorowstext">No data found</property></webpart>
You will need to update the property based on the name of that property.